Download 1. Basics of Geometry

Geometry
Authors
Victor Cifarelli, Andrew Gloag, Dan Greenberg, Jim Sconyers,
and Bill Zahner.
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Contents
1. Basics of Geometry.......................................................................................................................... 1
2. Reasoning and Proof..................................................................................................................... 67
3. Parallel and Perpendicular Lines................................................................................................ 117
4. Congruent Triangles.................................................................................................................... 197
5. Relationships Within Triangles................................................................................................... 265
6. Quadrilaterals............................................................................................................................... 331
7. Similarity....................................................................................................................................... 403
8. Right Triangle Trigonometry....................................................................................................... 459
9. Circles........................................................................................................................................... 523
10. Perimeter and Area...................................................................................................................... 615
11. Surface Area and Volume............................................................................................................ 665
v
1. Basics of Geometry
1.1 Points, Lines, and Planes
Learning Objectives
•
Understand the undefined terms point, line, and plane.
•
Understand defined terms, including space, segment, and ray.
•
Identify and apply basic postulates of points, lines, and planes.
•
Draw and label terms in a diagram.
Introduction
Welcome to the exciting world of geometry! Ahead of you lie many exciting discoveries that will help you
learn more about the world. Geometry is used in many areas—from art to science. For example, geometry
plays a key role in construction, fashion design, architecture, and computer graphics. This course focuses
on the main ideas of geometry that are the foundation of applications of geometry used everywhere. In this
chapter, you’ll study the basic elements of geometry. Later you will prove things about geometric shapes
using the vocabulary and ideas in this chapter—so make sure that you completely understand each of the
concepts presented here before moving on.
1.1.1 Undefined Terms
The three basic building blocks of geometry are points, lines, and planes. These are undefined terms.
While we cannot define these terms precisely, we can get an idea of what they are by looking at examples
and models.
A point is a location that has no size. To imagine a point, look at the period at the end of this sentence. Now
imagine that period getting smaller and smaller until it disappears. A point describes a location, such as the
location of the period, but a point has no size. We use dots (like periods) to represent points, but since the
dots themselves occupy space, these dots are not points—we only use dots as representations. Points are
labeled with a capital letter, as shown below.
A line is an infinite series of points in a row. A line does not occupy space, so to imagine a line you can
imagine the thinnest string you can think of, and shrink it until it occupies no space at all. A line has direction
and location, but still does not take up space. Lines are sometimes referred to by one italicized letter, but
they can also be identified by two points that are on the line. Lines are called one-dimensional, since they
have direction in one dimension.
1
The last undefined term is plane. You can think of a plane as a huge sheet of paper—so big that it goes on
forever! Imagine the paper as thin as possible, and extend it up, down, left, and right. Planes can be named
by letter, or by three points that lie in the plane. You already know one plane from your algebra class—the
-coordinate plane. Planes are called two-dimensional, since any point on a plane can be described by
two numbers, called coordinates, as you learned in algebra.
Notation Notes: As new terms are introduced, notation notes will help you learn how to write and say them.
1. Points are named using a single capital letter. The first image shows points
,
, and
.
2.
In the image of a line, the same line has several names. It can be called “line ”,
, or
. The
order of the letters does not matter when naming a line, so the same line can have many names. When
using two points to name a line, you must use the line symbol
above the letters.
3. Planes are named using a script (cursive) letter or by naming three points contained in the plane. The
illustrated plane can be called plane
or “the plane defined by points
,
, and
.”
Example 1
Which term best describes how San Diego, California, would be represented on a globe?
A. point
B. line
C. plane
A city is usually labeled with a dot, or point, on a globe. Though the city of San Diego occupies space, it is
reduced when placed on the globe. Its label is merely to show a location with reference to the other cities,
states, and countries on a globe. So, the correct answer is A.
Example 2
Which geometric object best models the surface of a movie screen?
(Source: http://commons.wikimedia.org/wiki/File:Airscreen.JPG; License: GNU Free Documentation)
A. point
B. line
2
C. plane
The surface of a movie screen extends in two dimensions: up and down and left to right. This description
most closely resembles a plane. So, the correct answer is C. Note that a plane is a model of the movie
screen, but the screen is not actually a plane. In geometry, planes extend infinitely, but the movie screen
does not.
1.1.2 Defined Terms
Now we can use point, line, and plane to define new terms. One word that has already been used is space.
Space is the set of all points expanding in three dimensions. Think back to the plane. It extended along two
different lines: up and down, and side to side. If we add a third direction, we have something that looks like
three-dimensional space. In algebra, the
plane is adapted to model space by adding a third axis
coming out of the page. The image below shows three perpendicular axes.
Points are said to be collinear if they lie along the same line. The picture below shows points
,
, and
are collinear. Point
is non-collinear with the other three since it does not lie in the same line.
Similarly, points and lines can be coplanar if they lie within the same plane. The diagram below shows two
lines (
and
non-coplanar with
) and one point (
and
) that are coplanar. It also shows line
and point
that are
.
3
A segment designates a portion of a line that has two endpoints. Segments are named by their endpoints.
Notation Notes: Just like lines, segments are written with two capital letters. For segments we use a bar
on top with no arrows. Segments can also be named in any order, so the segment above could be named
or
.
A ray is a portion of a line that has only one endpoint and extends infinitely in the other direction. Rays are
named by their endpoints and another point on the line. The endpoint always comes first in the name of a
ray.
Like segments, rays are named with two capital letters, and the symbol on top has one arrow. The ray is
always named with the endpoint first, so we would write
for the figure above.
An intersection is the point or set of points where lines, planes, segments, or rays cross each other. Intersections are very important since you can study the different regions they create.
In the image above,
is the point of intersection of
and
.
is the intersection of
.
Example 3
Which geometric object best models a straight road connecting two cities?
4
and
A. ray
B. line
C. segment
D. plane
Since the straight road connects two distinct points (cities), and we are interested in the section between
those two endpoints, the best term is segment. A segment has two endpoints. So, the correct answer is C.
Example 4
Which term best describes the relationship among the strings on a tennis racket?
(Source: http://commons.wikimedia.org/wiki/File:Tennis_Racket_and_Balls.jpg; License: GNU Free Documentation)
A. collinear
B. coplanar
C. non-collinear
D. non-coplanar
The strings of a tennis racket are like intersecting segments. They also are all located on the plane made
by the head of the racket. So, the best answer is B. Note that the strings are not really the same as segments
and they are not exactly coplanar, but we can still use the geometric model of a plane for the head of a
tennis racket, even if the model is not perfect.
1.1.3 Basic Postulates
Now that we have some basic vocabulary, we can talk about the rules of geometry. Logical systems like
geometry start with basic rules, and we call these basic rules postulates. We assume that a postulate is
true and by definition a postulate is a statement that cannot be proven.
A theorem is a statement that can be proven true using postulates, definitions, logic, and other theorems
we’ve already proven. Theorems are the “results” that are true given postulates and definitions. This section
introduces a few basic postulates that you must understand as you move on to learn other theorems.
The first of five postulates you will study in this lesson states that there is exactly one line through any two
points. You could test this postulate easily with a ruler, a piece of paper, and a pencil. Use your pencil to
draw two points anywhere on the piece of paper. Use your ruler to connect these two points. You’ll find that
there is only one possible straight line that goes through them.
5
Line Postulate: There is exactly one line through any two points.
Similarly, there is exactly one plane that contains any three non-collinear points. To illustrate this, ask three
friends to hold up the tips of their pencils, and try and lay a piece of paper on top of them. If your friends line
up their pencils (making the points collinear), there are an infinite number of possible planes. If one hand
moves out of line, however, there is only one plane that will contain all three points. The following image
shows five planes passing through three collinear points.
Plane Postulate: There is exactly one plane that contains any three non-collinear points.
If two coplanar points form a line, that line is also within the same plane.
Postulate: A line connecting points in a plane also lies within the plane.
6
Sometimes lines intersect and sometimes they do not. When two lines do intersect, the intersection will be
a single point. This postulate will be especially important when looking at angles and relationships between
lines. As an extension of this, the final postulate for this lesson states that when two planes intersect they
meet in a single line. The following diagrams show these relationships.
Postulate: The intersection of any two distinct lines will be a single point.
Postulate: The intersection of two planes is a line.
Example 5
How many non-collinear points are required to identify a plane?
A. 1
B. 2
7
C. 3
D. 4
The second postulate listed in this lesson states that you can identify a plane with three non-collinear points.
It is important to label them as non-collinear points since there are infinitely many planes that contain collinear
points. The answer is C.
Example 6
What geometric figure represents the intersection of the two planes below?
A. point
B. line
C. ray
D. plane
The fifth postulate presented in this lesson says that the intersection of two planes is a line. This makes
sense from the diagram as well. It is a series of points that extends infinitely in both directions, so it is definitely
a line. The answer is B.
1.1.4 Drawing and Labeling
It is important as you continue your study of geometry to practice drawing geometric shapes. When you
make geometric drawings, you need to be sure to follow the conventions of geometry so other people can
“read” your drawing. For example, if you draw a line, be sure to include arrows at both ends. With only one
arrow, it will appear as a ray, and without any arrows, people will assume that it is a line segment. Make
sure you label your points, lines, and planes clearly, and refer to them by name when writing explanations.
You will have many opportunities to hone your drawing skills throughout this geometry course.
Example 7
Draw and label the intersection of line
and ray
at point
.
To begin making this drawing, make a line with two points on it. Label the points
Next, add the ray. The ray will have an endpoint
ray and line will intersect at
8
, so point
and another point
should be on
and
.
. The description says that the
. It is not important from this description in
what direction
points.
The diagram above satisfies the conditions in the problem.
Lesson Summary
In this lesson, we explored points, lines, and planes. Specifically, we have learned:
•
The significance of the undefined terms point, line, and plane.
•
The significance of defined terms including space, segment, and ray.
•
How to identify and apply basic postulates of points, lines, and planes.
•
How to draw and label the terms you have studied in a diagram.
These skills are the building blocks of geometry. It is important to have these concepts solidified in your
mind as you explore other topics of geometry and mathematics.
Points to Consider
You can think of postulates as the basic rules of geometry. Other activities also have basic rules. For example,
in the game of soccer one of the basic rules is that players are not allowed to use their hands to move the
ball. How do the rules shape the way that the game is played? As you become more familiar with the geometric postulates, think about how the basic “rules of the game” in geometry determine what you can and
cannot do.
Now that you know some of the basics, we are going to look at how measurement is used in geometry.
1.1 Lesson Exercises
1. Draw an image showing all of the following:
a.
b.
c. Plane
intersecting
containing
but not
2. Name this line in three ways.
9
3. What is the best possible geometric model for a soccer field? Explain your answer.
(Source: http://commons.wikimedia.org/wiki/File:Coba-arena-ffm036.jpg, License: CCSA)
4. What type of geometric object is the intersection of a line and a plane? Draw your answer.
5. What type of geometric object is made by the intersection of three planes? Draw your answer.
6. What type of geometric object is made by the intersection of a sphere (a ball) and a plane? Draw your
answer.
7. Use geometric notation to explain this picture in as much detail as possible.
8. True or false: Any two distinct points are collinear. Justify your answer.
9. True or false: Any three distinct points determine a plane (or in other words, there is exactly one plane
passing through any three points). Justify your answer.
10. One of the statements in 8 or 9 is false. Rewrite the false statement to make it true.
Answers
1. Answers will vary, one possible example:
10
2.
,
,
(and other answers are possible).
3. A soccer field is like a plane since it is a flat two-dimensional surface.
4. A line and a plane intersect at a point. See the diagram for answer 1 for an illustration. If
tended to be a line, then the intersection of
and plane
would be point
were ex-
.
5. Three planes intersect at one point.
6. A circle.
11
7.
intersects
at point
.
8. True: The Line Postulate implies that you can always draw a line between any two points, so they must
be collinear.
9. False. Three collinear points could be at the intersection of an infinite number of planes. See the images
of intersecting planes for an illustration of this.
10. For 9 to be true, it should read: “Any three non-collinear points determine a plane.”
1.2 Segments and Distance
Learning Objectives
•
Measure distances using different tools.
•
Understand and apply the ruler postulate to measurement.
•
Understand and apply the segment addition postulate to measurement.
•
Use endpoints to identify distances on a coordinate grid.
Introduction
You have been using measurement for most of your life to understand quantities like weight, time, distance,
area, and volume. Any time you have cooked a meal, bought something, or played a sport, measurement
has played an important role. This lesson explores the postulates about measurement in geometry.
1.2.1 Measuring Distances
There are many different ways to identify measurements. This lesson will present some that may be familiar,
and probably a few that are new to you. Before we begin to examine distances, however, it is important to
identify the meaning of distance in the context of geometry. The distance between two points is defined by
the length of the line segment that connects them.
The most common way to measure distance is with a ruler. Also, distance can be estimated using scale on
a map. Practice this skill in the example below.
Notation Notes: When we name a segment we use the endpoints and and overbar with no arrows. For
example, “Segment AB” is written
. The length of a segment is named by giving the endpoints without
using an overline. For example, the length of
, which means the same as
is written
. In some books you may also see
, that is, it is the length of the segment with endpoints A and B.
Example 1
Use the scale to estimate the distance between Aaron’s house and Bijal’s house.
12
You need to find the distance between the two houses in the map. The scale shows a sample distance.
Use the scale to estimate the distance. You will find that approximately three segments the length of the
scale fit between the two points. Be careful—three is not the answer to this problem! As the scale shows
one unit equal to two miles, you must multiply three units by two miles.
The distance between the houses is about six miles.
You can also use estimation to identify measurements in other geometric figures. Remember to include
words like approximately, about, or estimation whenever you are finding an estimated answer.
1.2.2 Ruler Postulate
You have probably been using rulers to measure distances for a long time and you know that a ruler is a
tool with measurement markings.
Ruler Postulate: If you use a ruler to find the distance between two points, the distance will be the absolute
value of the difference between the numbers shown on the ruler.
The ruler postulate implies that you do not need to start measuring at the zero mark, as long as you use
subtraction to find the distance. Note, we say “absolute value” here since distances in geometry must always
be positive, and subtraction can yield a negative result.
Example 2
What distance is marked on the ruler in the diagram below?
The way to use the ruler is to find the absolute value of difference between the numbers shown. The line
segment spans from 3 cm to 8 cm.
The absolute value of the difference between the two numbers shown on the ruler is 5 cm. So, the line
segment is 5 cm long.
Example 3
13
Use a ruler to find the length of the line segment below.
Line up the endpoints with numbers on your ruler and find the absolute value of the difference between
those numbers. If you measure correctly, you will find that this segment measures 2.5 inches or 6.35 centimeters.
1.2.3 Segment Addition Postulate
Segment Addition Postulate: The measure of any line segment can be found by adding the measures of
the smaller segments that comprise it.
That may seem like a lot of confusing words, but the logic is quite simple. In the diagram below, if you add
the lengths of
and
, you will have found the length of
. In symbols,
.
Use the segment addition postulate to put distances together.
Example 4
The map below shows the distances between three collinear towns.
What is the distance between town 1 and town 3?
You can see that the distance between town 1 and town 2 is eight miles. You can also see that the distance
between town 2 and town 3 is five miles. Using the segment addition postulate, you can add these values
together to find the total distance between town 1 and town 3.
14
The total distance between town 1 and town 3 is 13 miles.
1.2.4 Distances on a Grid
In algebra you most likely worked with graphing lines in the
coordinate grid. Sometimes you can find
the distance between points on a coordinate grid using the values of the coordinates. If the two points line
up horizontally, look at the change of value in the
-coordinates. If the two points line up vertically, look at
the change of value in the
-coordinates. The change in value will show the distance between the points.
Remember to use absolute value, just like you did with the ruler. Later you will learn how to calculate distance
between points that do not line up horizontally or vertically.
Example 5
What is the distance between the two points shown below?
The two points shown on the grid are at (2,9) and (2,3). As these points line up vertically, look at the difference
in the
-values.
So, the distance between the two points is 6 units.
Example 6
What is the distance between the two points shown below?
The two points shown on the grid are at
and
These points line up horizontally, so look at
the difference in the
-values. Remember to take the absolute value of the difference between the values
to find the distance.
15
The distance between the two points is 7 units.
Lesson Summary
In this lesson, we explored segments and distances. Specifically, we have learned:
•
How to measure distances using many different tools.
•
To understand and apply the Ruler Postulate to measurement.
•
To understand and apply the Segment Addition Postulate to measurement.
•
How to use endpoints to identify distances on a coordinate grid.
These skills are useful whenever performing measurements or calculations in diagrams. Make sure that
you fully understand all concepts presented here before continuing in your study.
1.2 Lesson Exercises
1. Use a ruler to measure the length of
below.
2. According to the ruler in the following image, how long is the cockroach?
(Source: http://commons.wikimedia.org/wiki/File:Dubia-cockroach-female-near-ruler.jpg, License: Public
Domain)
3. The ruler postulate says that we could have measured the cockroach in 2 without using the 0 cm marker
as the starting point. If the same cockroach as the one in 2 had its head at 6.5 cm, where would its tail be
on the ruler?
16
4. Suppose
is exactly in the middle of
5. What is
in the diagram below?
and
What is
6. Find
in the diagram below:
7. What is the length of the segment connecting (-2,3) and (-2, -7) in the coordinate plane? Justify your answer.
8. True or false: If
cm and
9. True or false:
cm, then
cm.
.
10. One of the statements in 8 or 9 is false. Show why it is false, and then change the statement to make it
true.
Answers
1. Answers will vary depending on scaling when printed and the units you use.
2. 4.5 cm (yuck!).
3. The tail would be at either 11 cm or 2 cm, depending on which way the cockroach was facing.
4.
5.
.
.
6.
7. Since the points are at the same
-coordinates.
-coordinate, we find the absolute value of the difference of the
17
8. False.
9. True.
, but the absolute value sign makes them both positive.
10. Number 8 is false. See the diagram below for a counterexample. To make 8 true, we need to add
something like: “If points
,
, and
are collinear, and
is between
and
, then if
and
, then
.”
1.3 Rays and Angles
Learning Objectives
•
Understand and identify rays.
•
Understand and classify angles.
•
Understand and apply the protractor postulate.
•
Understand and apply the angle addition postulate.
Introduction
Now that you know about line segments and how to measure them, you can apply what you have learned
to other geometric figures. This lesson deals with rays and angles, and you can apply much of what you
have already learned. We will try to help you see the connections between the topics you study in this book
instead of dealing with them in isolation. This will give you a more well-rounded understanding of geometry
and make you a better problem solver.
1.3.1 Rays
A ray is a part of a line with exactly one endpoint that extends infinitely in one direction. Rays are named by
their endpoint and a point on the ray.
The ray above is called
. The first letter in the ray’s name is always the endpoint of the ray, it doesn’t
matter which direction the ray points.
Rays can represent a number of different objects in the real world. For example, the beam of light extending
from a flashlight that continues forever in one direction is a ray. The flashlight would be the endpoint of the
ray, and the light continues as far as you can imagine so it is the infinitely long part of the ray. Are there
other real-life objects that can be represented as rays?
Example 1
18
Which of the figures below shows
Remember that a ray has one endpoint and extends infinitely in one direction. Choice A is a line segment
since it has two endpoints. Choice B has one endpoint and extends infinitely in one direction, so it is a ray.
Choice C has no endpoints and extends infinitely in two directions—it is a line. Choice D also shows a ray
with endpoint
Since we need to identify
with endpoint
, we know that choice B is correct.
Example 2
Use this space to draw
.
Remember that you are not expected to be an artist. In geometry, you simply need to draw figures that
accurately represent the terms in question. This problem asks you to draw a ray. Begin with a line segment.
Use your ruler to draw a straight line segment of any length.
Now draw an endpoint on one end and an arrow on the other.
Finally, label the endpoint
The diagram above shows
and another point on the ray
.
.
1.3.2 Angles
An angle is formed when two rays share a common endpoint. That common endpoint is called the vertex
and the two rays are called the sides of the angle. In the diagram below,
, or
for short. The symbol
is used for naming angles.
and
form an angle,
The same basic definition for angle also holds when lines, segments, or rays intersect.
19
Notation Notes: 1) Angles can be named by a number, a single letter at the vertex, or by the three points
that form the angle. When an angle is named with three letters, the middle letter will always be the vertex
of the angle. In the diagram above, the angle can be written
, or
, or
. You can use
one letter to name this angle since point
is the vertex and there is only one angle at point
. 2) If two
or more angles share the same vertex, you MUST use three letters to name the angle. For example, in the
image below it is unclear which angle is referred to by
. To talk about the angle with one arc, you would
write
. For the angle with two arcs, you’d write
.
We use a ruler to measure segments by their length. But how do we measure an angle? The length of the
sides does not change how wide an angle is “open.” Instead of using length, the size of an angle is measured
by the amount of rotation from one side to another. By definition, a full turn is defined as 360 degrees. Use
the symbol ° for degrees. You may have heard “360” used as slang for a “full circle” turn, and this expression
comes from the fact that a full rotation is 360°.
The angle that is made by rotating through one-fourth of a full turn is very special. It measures
and we call this a right angle. Right angles are easy to identify, as they look like the corners
of most buildings, or a corner of a piece of paper.
A right angle measures exactly 90º.
Right angles are usually marked with a small square. When two lines, two segments, or two rays intersect
at a right angle, we say that they are perpendicular. The symbol
is used for two perpendicular lines.
20
An acute angle measures between 0º and 90º.
An obtuse angle measures between 90º and 180º.
A straight angle measures exactly 180º. These are easy to spot since they look like straight lines.
You can use this information to classify any angle you see.
Example 3
What is the name and classification of the angle below?
21
Begin by naming this angle. It has three points labeled and the vertex is
. So, the angle will be named
or just
. For the classification, compare the angle to a right angle.
opens wider than
a right angle, and less than a straight angle. So, it is obtuse.
Example 4
What term best describes the angle formed by Clinton and Reeve streets on the map below?
The intersecting streets form a right angle. It is a square corner, so it measures 90º.
1.3.3 Protractor Postulate
In the last lesson, you studied the ruler postulate. In this lesson, we’ll explore the Protractor Postulate. As
you can guess, it is similar to the ruler postulate, but applied to angles instead of line segments. A protractor
is a half-circle measuring device with angle measures marked for each degree. You measure angles with
a protractor by lining up the vertex of the angle on the center of the protractor and then using the protractor
postulate (see below). Be careful though, most protractors have two sets of measurements—one opening
clockwise and one opening counterclockwise Make sure you use the same scale when reading the measures
of the angle.
Protractor Postulate: For every angle there is a number between 0 and 180 that is the measure of the
angle in degrees. You can use a protractor to measure an angle by aligning the center of the protractor on
the vertex of the angle. The angle’s measure is then the absolute value of the difference of the numbers
shown on the protractor where the sides of the angle intersect the protractor.
It is probably easier to understand this postulate by looking at an example. The basic idea is that you do
not need to start measuring an angle at the zero mark, as long as you find the absolute value of the difference
22
of the two measurements. Of course, starting with one side at zero is usually easier. Examples 5 and 6
show how to use a protractor to measure angles.
Notation Note: When we talk about the measure of an angle, we use the symbols
. So for example,
if we used a protractor to measure
in example 3 and we found that it measured 120º, we could
write
.
Example 5
What is the measure of the angle shown below?
This angle is lined up with a protractor at 0º, so you can simply read the final number on the protractor itself.
Remember you can check that you are using the correct scale by making sure your answer fits your angle.
If the angle is acute, the measure of the angle should be less than 90º. If it is obtuse, the measure will be
greater than 90º. In this case, the angle is acute, so its measure is 50º.
Example 6
What is the measure of the angle shown below?
This angle is not lined up with the zero mark on the protractor, so you will have to use subtraction to find
its measure.
Using the inner scale, we get
.
Using the outer scale,
.
Notice that it does not matter which scale you use. The measure of the angle is 125º.
Example 7
Use a protractor to measure
below.
23
You can either line it up with zero, or line it up with another number and find the absolute value of the differences of the angle measures at the endpoints. Either way, the result is 100º. The angle measures 100º.
1.3.4 Angle Addition Postulate
You have already encountered the ruler postulate and the protractor postulate. There is also a postulate
about angles that is similar to the Segment Addition Postulate.
Angle Addition Postulate: The measure of any angle can be found by adding the measures of the smaller
angles that comprise it. In the diagram below, if you add
and
, you will have found
.
Use this postulate just as you did the segment addition postulate to identify the way different angles combine.
Example 8
What is
in the diagram below?
You can see that
is 15º. You can also see that
tulate, you can add these values together to find the total
24
is 30º. Using the angle addition pos.
So,
is 45º.
Example 9
What is
To find
in the diagram below given
, you must subtract
So,
and
from
?
.
.
Lesson Summary
In this lesson, we explored rays and angles. Specifically, we have learned:
•
To understand and identify rays.
•
To understand and classify angles.
•
To understand and apply the Protractor Postulate.
•
To understand and apply the Angle Addition Postulate.
These skills are useful whenever studying rays and angles. Make sure that you fully understand all concepts
presented here before continuing in your study.
1.3 Lesson Exercises
Use this diagram for questions 1-4.
1. Give two possible names for the ray in the diagram.
2. Give four possible names for the line in the diagram.
3. Name an acute angle in the diagram.
25
4. Name an obtuse angle in the diagram.
5. Name a straight angle in the diagram.
6. Which angle can be named using only one letter?
7. Explain why it is okay to name some angles with only one angle, but with other angles this is not okay.
8. Use a protractor to find
below:
(Source: http://commons.wikimedia.org/wiki/File:Protractor.jpg; License: GNU Free Documentation)
and
9. Given
, find
.
10. True or false: Adding two acute angles will result in an obtuse angle. If false, provide a counterexample.
Answers
1.
or
2.
,
,
, or
are four possible answers. There are more (how many?)
3.
4.
or
or
5.
6. Angle
7. If there is more than one angle at a given vertex, then you must use three letters to name the angle. If
there is only one angle at a vertex (as in angle
above) then it is permissible to name the angle with one
letter.
26
8.
9.
.
10. False. For a counterexample, suppose two acute angles measure
and
angles is
, but
is still acute. See the diagram for a counterexample:
, then the sum of those
1.4 Segments and Angles
Learning Objectives
•
Understand and identify congruent line segments.
•
Identify the midpoint of line segments.
•
Identify the bisector of a line segment.
•
Understand and identify congruent angles.
•
Understand and apply the Angle Bisector Postulate.
Introduction
Now that you have a better understanding of segments, angles, rays, and other basic geometric shapes,
we can study the ways in which they can be divided. Any time you come across a segment or an angle,
there are different ways to separate it into parts.
1.4.1 Congruent Line Segments
One of the most important words in geometry is congruent. This term refers to geometric objects that have
exactly the same size and shape. Two segments are congruent if they have the same length.
Notation Notes:
1. When two things are congruent we use the symbol
we would write
. For example if
is congruent to
, then
.
2. When we draw congruent segments, we use tic marks to show that two segments are congruent.
27
3. If there are multiple pairs of congruent segments (which are not congruent to each other) in the same
picture, use two tic marks for the second set of congruent segments, three for the third set, and so on.
See the two following illustrations.
Recall that the length of segment
can be written in two ways:
or simply
. This might be
a little confusing at first, but it will make sense as you use this notation more and more. Let’s say we used
a ruler and measured
or
cm.
If we know that
and we saw that it had a length of 5 cm. Then we could write
, then we can write
or simply
cm,
.
You can prove two segments are congruent in a number of ways. You can measure them to find their lengths
using any units of measurement—the units do not matter as long as you use the same units for both measurements. Or, if the segments are drawn in the
plane, you can also find their lengths on the coordinate
grid. Later in the course you will learn other ways to prove two segments are congruent.
Example 1
Henrietta drew a line segment on a coordinate grid as shown below.
She wants to draw another segment congruent to the first that begins at (-1,1) and travels straight up (that
is, in the
direction). What will be the coordinates of its second endpoint?
You will have to solve this problem in stages. The first step is to identify the length of the segment drawn
onto the grid. It begins at (2,3) and ends at (6,3). So, its length is 4 units.
28
The next step is to draw the second segment. Use a pencil to create the segment according to the specifications in the problem. You know that the segment needs to be congruent to the first, so it will be 4 units
long. The problem also states that it travels straight up from the point (-1,1). Draw in the point at (-1,1) and
make a line segment 4 units long that travels straight up.
Now that you have drawn in the new segment, use the grid to identify the new endpoint. It has an
dinate of -1 and a
-coordinate of 5. So, its coordinates are (-1,5).
-coor-
1.4.2 Segment Midpoints
Now that you understand congruent segments, there are a number of new terms and types of figures you
can explore. A segment midpoint is a point on a line segment that divides the segment into two congruent
segments. So, each segment between the midpoint and an endpoint will have the same length. In the diagram
below, point
is the midpoint of segment
since
is congruent to
.
There is even a special postulate dedicated to midpoints.
Segment Midpoint Postulate: Any line segment will have exactly one midpoint—no more, and no less.
Example 2
Nandi and Arshad measure and find that their houses are 10 miles apart. If they agree to meet at the midpoint
between their two houses, how far will each of them travel?
The easiest way to find the distance to the midpoint of the imagined segment connecting their houses is to
divide the length by 2.
29
So, each person will travel five miles to meet at the midpoint between Nandi’s and Arshad’s houses.
1.4.3 Segment Bisectors
Now that you know how to find midpoints of line segments, you can explore segment bisectors. A bisector
is a line, segment, or ray that passes through a midpoint of another segment. You probably know that the
prefix “bi” means two (think about the two wheels of a bicycle). So, a bisector cuts a line segment into two
congruent parts.
Example 3
Use a ruler to draw a bisector of the segment below.
The first step in identifying a bisector is finding the midpoint. Measure the line segment to find that it is 4
cm long. To find the midpoint, divide this distance by 2.
So, the midpoint will be 2 cm from either endpoint on the segment. Measure 2 cm from an endpoint and
draw the midpoint.
To complete the problem, draw a line segment that passes through the midpoint. It doesn’t matter what
angle this segment travels on. As long as it passes through the midpoint, it is a bisector.
1.4.4 Congruent Angles
You already know that congruent line segments have exactly the same length. You can also apply the
concept of congruence to other geometric figures. When angles are congruent, they have exactly the same
measure. They may point in different directions, have different side lengths, have different names or other
attributes, but their measures will be equal.
Notation Notes:
1. When writing that two angles are congruent, we use the congruent symbol:
Alternatively, the symbol
refers to the measure of
, so we could write
and that has the same meaning as
. You may notice
then, that numbers (such as measurements) are equal while objects (such as angles and segments)
are congruent.
2. When drawing congruent angles, you use an arc in the middle of the angle to show that two angles are
congruent. If two different pairs of angles are congruent, use one set of arcs for one pair, then two for
the next pair and so on.
30
Use algebra to find a way to solve the problem below using this information.
Example 4
The two angles shown below are congruent.
What is the measure of each angle?
This problem combines issues of both algebra and geometry, so make sure you set up the problem correctly.
It is given that the two angles are congruent, so they must have the same measurements. Therefore, you
can set up an equation in which the expressions representing the angle measures are equal to each other.
Now that you have an equation with one variable, you can solve for the value of
So, the value of
is 8. You are not done, however. Use this value of
angles in the problem.
.
to find the measure of one of the
31
Finally, we know
, so both of the angles measure 47º.
1.4.5 Angle Bisectors
If a segment bisector divides a segment into two congruent parts, you can probably guess what an angle
bisector is. An angle bisector divides an angle into two congruent angles, each having a measure exactly
half of the original angle.
Angle Bisector Postulate: Every angle has exactly one bisector.
Example 5
The angle below measures 136º.
If a bisector is drawn in this angle, what will be the measure of the new angles formed?
This is similar to the problem about the midpoint between the two houses. To find the measurements of the
smaller angles once a bisector is drawn, divide the original angle measure by 2:
So, each of the newly formed angles would measure 68º when the 136º angle is bisected.
Lesson Summary
In this lesson, we explored segments and angles. Specifically, we have learned:
32
•
How to understand and identify congruent line segments.
•
How to identify the midpoint of line segments.
•
How to identify the bisector of a line segment.
•
How to understand and identify congruent angles.
•
How to understand and apply the Angle Bisector Postulate.
These skills are useful whenever performing measurements or calculations in diagrams. Make sure that
you fully understand all concepts presented here before continuing in your study.
1.4 Lesson Exercises
1. Copy the figure below and label it with the following information:
a.
b.
c.
2. Sketch and label an angle bisector
3. If we know that
of
below.
, what is
Use the following diagram of rectangle
for questions 4-10. (For these problems you can assume
that opposite sides of a rectangle are congruent—later you will prove this is true.)
Given that
is the midpoint of
and
, find the following lengths:
4.
33
5.
6.
7.
8.
9.
10.
11. How many copies of
Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
34
can fit inside rectangle
10.
11. 8
1.5 Angle Pairs
Learning Objectives
•
Understand and identify complementary angles.
•
Understand and identify supplementary angles.
•
Understand and utilize the Linear Pair Postulate.
•
Understand and identify vertical angles.
Introduction
In this lesson you will learn about special angle pairs and prove the vertical angles theorem, one of the most
useful theorems in geometry.
1.5.1 Complementary Angles
A pair of angles are Complementary angles if the sum of their measures is 90º.
Complementary angles do not have to be congruent to each other. Rather, their only defining quality is that
the sum of their measures is equal to the measure of a right angle: 90º. If the outer rays of two adjacent
angles form a right angle, then the angles are complementary.
Example 1
The two angles below are complementary.
. What is the value of
?
Since you know that the two angles must sum to 90º, you can create an equation. Then solve for the variable.
In this case, the variable is
.
35
Thus, the value of
is 56º.
Example 2
The two angles below are complementary. What is the measure of each angle?
This problem is a bit more complicated than the first example. However, the concepts are the same. If you
add the two angles together, the sum will be 90º. So, you can set up an algebraic equation with the values
presented.
The best way to solve this problem is to solve the equation above for
. Then, you must substitute the
value for back into the original expressions to find the value of each angle.
The value of
is 5. Now substitute this value back into the expressions to find the measures of the two
angles in the diagram.
and
verifying if they are complementary.
. You can check to make sure these numbers are accurate by
Since these two angle measures sum to 90º, they are complementary.
1.5.2 Supplementary Angles
Two angles are supplementary if their measures sum to 180º.
36
Just like complementary angles, supplementary angles need not be congruent, or even touching. Their
defining quality is that when their measures are added together, the sum is 180º. You can use this information
just as you did with complementary angles to solve different types of problems.
Example 3
The two angles below are supplementary. If
, what is
?
This process is very straightforward. Since you know that the two angles must sum to 180º, you can create
an equation. Use a variable for the unknown angle measure and then solve for the variable. In this case,
let’s call
.
So, the measure of
and thus
.
Example 4
What is the measure of two congruent, supplementary angles?
There is no diagram to help you visualize this scenario, so you’ll have to imagine the angles (or even better,
draw it yourself by translating the words into a picture!). Two supplementary angles must sum to 180º.
Congruent angles must have the same measure. So, you need to find two congruent angles that are supplementary. You can divide 180º by two to find the value of each angle.
Each congruent, supplementary angle will measure 90º. In other words, they will be right angles.
1.5.3 Linear Pairs
Before we talk about a special pair of angles called linear pairs, we need to define adjacent angles. Two
angles are adjacent if they share the same vertex and one side, but they do not overlap. In the diagram
below,
and
are adjacent.
37
However,
and
interior of the angle).
are not adjacent since they overlap (i.e. they share common points in the
Now we are ready to talk about linear pairs. A linear pair is two angles that are adjacent and whose noncommon sides form a straight line. In the diagram below,
and
are a linear pair. Note
that
is a line.
Linear pairs are so important in geometry that they have their own postulate.
Linear Pair Postulate: If two angles are a linear pair, then they are supplementary.
Example 5
The two angles below form a linear pair. What is the value of each angle?
If you add the two angles, the sum will be 180º. So, you can set up an algebraic equation with the values
presented.
The best way to solve this problem is to solve the equation above for
for back into the original expressions to find the value of each angle.
38
. Then, you must plug the value
The value of is 9. Now substitute this value back into the expressions to determine the measures of the
two angles in the diagram.
The two angles in the diagram measure 27º and 153º. You can check to make sure these numbers are
accurate by verifying if they are supplementary.
1.5.4 Vertical Angles
Now that you understand supplementary and complementary angles, you can examine more complicated
situations. Special angle relationships are formed when two lines intersect, and you can use your knowledge
of linear pairs of angles to explore each angle further.
Vertical angles are defined as two non-adjacent angles formed by intersecting lines. In the diagram below,
and
are vertical angles. Also,
and
are vertical angles.
Suppose that you know
other angles. For example,
and
, subtract 100º from 180º.
. You can use that information to find the measurement of all the
must be supplementary since they are a linear pair. So, to find
39
So
measures 80º. Knowing that angles 2 and 3 are also supplementary means that
,
since the sum of 100º and 80º is 180º. If angle 3 measures 100º, then the measure of angle 4 must be 80º,
since 3 and 4 are also supplementary. Notice that angles 1 and 3 are congruent (100º) and 2 and 4 are
congruent (80º).
The vertical angles theorem states that if two angles are vertical angles then they are congruent.
We can prove the vertical angles theorem using a process just like the one we used above. There was
nothing special about the given measure of
. Here is proof that vertical angles will always be congruent:
Since
and
form a linear pair, we know that they are supplementary:
the same reason,
and
are supplementary:
write
. Finally, subtracting
Or, by the definition of congruent angles,
.
. For
. Using a substitution, we can
on both sides yields
.
Use your knowledge of vertical angles to solve the following problem.
Example 6
What is
in the diagram below?
Using your knowledge of intersecting lines, you can identify that
is vertical to the angle marked
18º. Since vertical angles are congruent, they will have the same measure. So,
is also equal to
18º.
Lesson Summary
In this lesson, we explored angle pairs. Specifically, we have learned:
40
•
How to understand and identify complementary angles.
•
How to understand and identify supplementary angles.
•
How to understand and utilize the Linear Pair Postulate.
•
How to understand and identify vertical angles.
The relationships between different angles are used in almost every type of geometric application. Make
sure that these concepts are retained as you progress in your studies.
1.5 Lesson Exercises
1. Find the measure of the angle complementary to
if
o
a. 45
o
b. 82
o
c. 19
d. z
o
2. Find the measure of the angle supplementary to
if
o
a. 45
o
b. 118
o
c. 32
d. x
o
3. Find
4. Given
and
.
, Find
.
Use the diagram below for exercises 5 and 6. Note that
.
41
5. Identify each of the following (there may be more than one correct answer for some of these questions).
a. Name one pair of vertical angles.
b. Name one linear pair of angles.
c. Name two complementary angles.
d. Nam two supplementary angles.
, find
6. Given that
a.
.
b.
.
c.
.
d.
.
Answers
o
o
o
o
1. a) 45 , b) 8 , c) 81 , d) (90 - z)
o
o
o
o
2. a) 135 , b) 62 , c) 148 , d) (180 - x)
3.
,
4.
5. a)
and
and
and
(or
also works); c)
also works).
o
o
o
and
and
also works); b)
; d) same as (b)...
o
6. a) 27 , b) 90 , c) 63 , d) 117
1.6 Classifying Triangles
Learning Objectives
42
•
Define triangles.
•
Classify triangles as acute, right, obtuse, or equiangular.
and
and
(or
(or
•
Classify triangles as scalene, isosceles, or equilateral.
Introduction
By this point, you should be able to readily identify many different types of geometric objects. You have
learned about lines, segments, rays, planes, as well as basic relationships between many of these figures.
Everything you have learned up to this point is necessary to explore the classifications and properties of
different types of shapes. The next two sections focus on two-dimensional shapes—shapes that lie in one
plane. As you learn about polygons, use what you know about measurement and angle relationships in
these sections.
1.6.1 Defining Triangles
The first shape to examine is the triangle. Though you have probably heard of triangles before, it is helpful
to review the formal definition. A triangle is any closed figure made by three line segments intersecting at
their endpoints. Every triangle has three vertices (points at which the segments meet), three sides (the
segments themselves), and three interior angles (formed at each vertex). All of the following shapes are
triangles.
You may have learned in the past that the sum of the interior angles in a triangle is always 180º. Later we
will prove this property, but for now you can use this fact to find missing angles. Other important properties
of triangles will be explored in later chapters.
Example 1
Which of the figures below are not triangles?
To solve this problem, you must carefully analyze the four shapes in the answer choices. Remember that
a triangle has three sides, three vertices, and three interior angles. Choice A fits this description, so it is a
triangle. Choice B has one curved side, so its sides are not exclusively line segments. Choice C is also a
triangle. Choice D, however, is not a closed shape. Therefore, it is not a triangle. Choices B and D are not
triangles.
Example 2
How many triangles are in the diagram below?
43
To solve this problem, you must carefully count the triangles of different size. Begin with the smallest triangles.
There are 16 small triangles.
Now count the triangles that are formed by four of the smaller triangles, like the one below.
There are a total of seven triangles of this size, if you remember to count the inverted one in the center of
the diagram.
Next, count the triangles that are formed by nine of the smaller triangles. There are three of these triangles.
And finally, there is one triangle formed by 16 smaller triangles.
Now, add these numbers together.
16 + 7 + 3 + 1 = 27
So, there are a total of 27 triangles in the figure shown.
1.6.2 Classifications by Angles
Earlier in this chapter, you learned how to classify angles as acute, obtuse, or right. Now that you know how
to identify triangles, we can separate them into classifications as well. One way to classify a triangle is by
the measure of its angles. In any triangle, two of the angles will always be acute. This is necessary to keep
the total sum of the interior angles at 180º. The third angle, however, can be acute, obtuse, or right.
This is how triangles are classified. If a triangle has one right angle, it is called a right triangle.
44
If a triangle has one obtuse angle, it is called an obtuse triangle.
If all of the angles are acute, it is called an acute triangle.
The last type of triangle classifications by angles occurs when all angles are congruent. This triangle is called
an equiangular triangle.
Example 3
Which term best describes
below?
45
The triangle in the diagram has two acute angles. But,
so
is an obtuse angle.
If the angle measure were not given you could check this using the corner of a piece of notebook paper or
by measuring the angle with a protractor. An obtuse angle will be greater than 90º (the square corner of a
paper) and less than 180º (a straight line). Since one angle in the triangle above is obtuse, it is an obtuse
triangle.
1.6.3 Classifying by Side Lengths
There are more types of triangle classes that are not based on angle measure. Instead, these classifications
have to do with the sides of the triangle and their relationships to each other. When a triangle has all sides
of different length, it is called a scalene triangle.
When at least two sides of a triangle are congruent, the triangle is said to be an isosceles triangle.
Finally, when a triangle has sides that are all congruent, it is called an equilateral triangle. Note that by the
definitions, an equilateral triangle is also an isosceles triangle.
Example 4
Which term best describes the triangle below?
46
To classify the triangle by side lengths, you have to examine the relationships between the sides. Two of
the sides in this triangle are congruent, so it is an isosceles triangle. The correct answer is B.
Lesson Summary
In this lesson, we explored triangles and their classifications. Specifically, we have learned:
•
How to define triangles.
•
How to classify triangles as acute, right, obtuse, or equiangular.
•
How to classify triangles as scalene, isosceles, or equilateral.
These terms or concepts are important in many different types of geometric practice. It is important to have
these concepts solidified in your mind as you explore other topics of geometry and mathematics.
1.6 Lesson Exercises
Exercises 1-5: Classify each triangle by its sides and by its angles. If you do not have enough information
to make a classification, write “not enough information.”
47
6. Sketch an equiangular triangle. What must be true about the sides?
7. Sketch an obtuse isosceles triangle.
8. True or false: A right triangle can be scalene.
9. True or false: An obtuse triangle can have more than one obtuse angle.
10. One of the answers in 8 or 9 is false. Sketch an illustration to show why it is false, and change the false
statement to make it true.
Answers
1) A is an acute scalene triangle.
2) B is an equilateral triangle.
3) C is a right isosceles triangle.
4) D is a scalene triangle. Since we don’t know anything about the angles, we cannot assume it is a right
o
triangle, even though one of the angles looks like it may be 90 .
5) E is an obtuse scalene triangle.
6) If a triangle is equiangular then it is also equilateral, so the sides are all congruent.
7) Sketch below:
48
8) True.
9) False.
10) 9 is false since the three sides would not make a triangle. To make the statement true, it should say:
“An obtuse triangle has exactly one obtuse angle.”
1.7 Classifying Polygons
Learning Objectives
•
Define polygons.
•
Understand the difference between convex and concave polygons.
•
Classify polygons by number of sides.
•
Use the distance formula to find side lengths on a coordinate grid.
Introduction
As you progress in your studies of geometry, you can examine different types of shapes. In the last lesson,
you studied the triangle, and different ways to classify triangles. This lesson presents other shapes, called
polygons. There are many different ways to classify and analyze these shapes. Practice these classification
procedures frequently and they will get easier and easier.
1.7.1 Defining Polygons
Now that you know what a triangle is, you can learn about other types of shapes. Triangles belong to a
larger group of shapes called polygons. A polygon is any closed planar figure that is made entirely of line
segments that intersect at their endpoints. Polygons can have any number of sides and angles, but the sides
can never be curved.
The segments are called the sides of the polygons, and the points where the segments intersect are called
vertices. Note that the singular of vertices is vertex.
49
The easiest way to identify a polygon is to look for a closed figure with no curved sides. If there is any curvature in a shape, it cannot be a polygon. Also, the points of a polygon must all lie within the same plane
(or it wouldn’t be two-dimensional).
Example 1
Which of the figures below is a polygon?
The easiest way to identify the polygon is to identify which shapes are not polygons. Choices B and C each
have at least one curved side. So they cannot be polygons. Choice D has all straight sides, but one of the
vertices is not at the endpoints of the two adjacent sides, so it is not a polygon. Choice A is composed entirely
of line segments that intersect at their endpoints. So, it is a polygon. The correct answer is A.
Example 2
Which of the figures below is not a polygon?
All four of the shapes are composed of line segments, so you cannot eliminate any choices based on that
criteria alone. Notice that choices A, B, and D have points that all lie within the same plane. Choice C is a
three-dimensional shape, so it does not lie within one plane. So it is not a polygon. The correct answer is
C.
1.7.2 Convex and Concave Polygons
Now that you know how to identify polygons, you can begin to practice classifying them. The first type of
classification to learn is whether a polygon is convex or concave. Think of the term concave as referring
to a cave, or an interior space. A concave polygon has a section that “points inward” toward the middle of
the shape. In any concave polygon, there are at least two vertices that can be connected without passing
through the interior of the shape. The polygon below is concave and demonstrates this property.
50
A convex polygon does not share this property. Any time you connect the vertices of a convex polygon, the
segments between nonadjacent vertices will travel through the interior of the shape. Lines segments that
connect to vertices traveling only on the interior of the shape are called diagonals.
Example 3
Identify whether the shapes below are convex or concave.
To solve this problem, connect the vertices to see if the segments pass through the interior or exterior of
the shape.
A. The segments go through the interior.
Therefore, the polygon is convex.
B. The segments go through the exterior.
51
Therefore, the polygon is concave.
C. One of the segments goes through the exterior.
Thus, the polygon is concave.
1.7.3 Classifying Polygons
The most common way to classify a polygon is by the number of sides. Regardless of whether the polygon
is convex or concave, it can be named by the number of sides. The prefix in each name reveals the number
of sides. The chart below shows names and samples of polygons.
52
Polygon Name
Number of Sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Heptagon
7
Sample Drawings
Octagon
8
Nonagon
9
Decagon
10
Undecagon or hen- 11
decagon (there is
some debate!)
Dodecagon
12
n - gon
n (where
)
Practice using these polygon names with the appropriate prefixes. The more you practice, the more you will
remember.
Example 4
53
Name the three polygons below by their number of sides.
A. This shape has seven sides, so it is a heptagon.
B. This shape has five sides, so it is a pentagon.
C. This shape has ten sides, so it is a decagon.
1.7.4 Using the Distance Formula on Polygons
You can use the distance formula to find the lengths of sides of polygons if they are on a coordinate grid.
Remember to carefully assign the values to the variables to ensure accuracy. Recall from algebra that you
and
can find the distance between points
using the following formula.
Example 5
A quadrilateral has been drawn on the coordinate grid below.
What is the length of segment
?
Use the distance formula to solve this problem. The endpoints of
for
, 9 for
, 4 for
, and 1 for
. Then we have:
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are (-3,9) and (4,1). Substitute -3
So the distance between points
and
is
, or about 10.63 units.
Lesson Summary
In this lesson, we explored polygons. Specifically, we have learned:
•
How to define polygons.
•
How to understand the difference between convex and concave polygons.
•
How to classify polygons by number of sides.
•
How to use the distance formula to find side lengths on a coordinate grid.
Polygons are important geometric shapes, and there are many different types of questions that involve them.
Polygons are important aspects of architecture and design and appear constantly in nature. Notice the
polygons you see every day when you look at buildings, chopped vegetables, and even bookshelves. Make
sure you practice the classifications of different polygons so that you can name them easily.
1.7 Lesson Exercises
For exercises 1-5, name each polygon in as much detail as possible.
6. Explain why the following figures are NOT polygons:
55
7. How many diagonals can you draw from one vertex of a pentagon? Draw a sketch of your answer.
8. How many diagonals can you draw from one vertex of an octagon? Draw a sketch of your answer.
9. How many diagonals can you draw from one vertex of a dodecagon?
10. Use your answers to 7, 8, and 9 and try more examples if necessary to answer the question: How many
diagonals can you draw from one vertex of an
-gon?
Answers
1. This is a convex pentagon.
2. Concave octagon.
3. Concave 17-gon (note that the number of sides is equal to the number of vertices, so it may be easier to
count the points [vertices] instead of the sides).
4. Concave decagon.
5. Convex quadrilateral.
6. A is not a polygon since the two sides do not meet at a vertex; B is not a polygon since one side is curved;
C is not a polygon since it is not enclosed.
7. The answer is 2.
8. The answer is 5.
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9. A dodecagon has twelve sides, so you can draw nine diagonals from one vertex.
10. Use this table to answer question 10,
Sides
Diagonals from One Vertex
...
...
To see the pattern, try adding a “process” column that takes you from the left column to the right side.
Sides
...
Process
Diagonals from One Vertex
...
Notice that we subtract 3 from each number on the left to arrive at the number in the right column. So, if the
number in the left column is
(standing for some unknown number), then the number in the right column
57
is
.
1.8 Problem Solving in Geometry
Learning Objectives
•
Read and understand given problem situations.
•
Use multiple representations to restate problem situations.
•
Identify problem-solving plans.
•
Solve real-world problems using planning strategies.
Introduction
One of the most important things we hope you will learn in school is how to solve problems. In real life,
problem solving is not usually as clear as it is in school. Often, performing a calculation or measurement
can be a simple task. Knowing what to measure or solve for can be the greatest challenge in solving problems.
This lesson helps you develop the skills needed to become a good problem solver.
1.8.1 Understanding Problem Situations
The first step whenever you approach a complicated problem is to simplify the problem. That means identifying the necessary information, and finding the desired value. Begin by asking yourself the simple question:
What is this problem asking for?
If the problem had to ask you only one question, what would it be? This helps you identify how you should
respond in the end.
Next, you have to find the information you need to solve the problem. Ask yourself another question: What
do I need to know to find the answer?
This question will help you sift through information that may be helpful with this problem.
Use these basic questions to simplify the following problem. Don’t try to solve it yet, just begin this process
with questioning.
Example 1
Ehab drew a rectangle
in the diagonal
on the chalkboard.
was
, what will be its length?
Begin to understand this problem by asking yourself two questions:
1. What is the problem asking for?
The question asks for the length of diagonal
.
2. What do I need to know to find the answer?
You need to know three things:
•
58
The angles of a rectangle are all equal to
.
and
was
. If Ehab draws
•
The lengths of the sides of the rectangle are 8 cm and 6 cm.
•
The Pythagorean Theorem can be used to find the third side of a right triangle.
Answering these questions is the first step to success with this problem.
1.8.2 Drawing Representations
Up to this point, the analysis of the sample problem has dealt with words alone. It is important to distill the
basic information from the problem, but there are different ways to proceed from here. Often, visual representations can be very helpful in understanding problems. Make a simple drawing that represents what is
being discussed. For example, a tray with six cookies could be represented by the diagram below.
The drawing takes only seconds to create, but it could help you visualize important information. Remember
that there are many different ways to display information. Look at the way a line segment six inches long is
displayed below.
When you approach a problem, think about how you can represent the information in the most useful way.
Continue your work on the sample problem by making drawings.
Let’s return to that example.
Example 1 (Repeated)
59
Ehab drew a rectangle
in the diagonal
on the chalkboard.
was
and
was
. If Ehab draws
, what will be its length?
Think about the different ways in which you could draw the information in this problem. The simplest idea
is to draw a labeled rectangle. Be sure to label your drawing with information from the problem. This includes
the names of the vertices as well as the side lengths.
As in most situations that you will encounter, there is more than one correct way to draw this shape. Two
more possibilities follow.
The first example above shows the internal structure of the rectangle, as it is divided into square centimeters.
The second example shows the rectangle situated on a coordinate grid. Notice that we rotated the figure
by 90° in the second picture. This is fine as long as it was drawn maintaining side lengths. One implication
of putting the figure on the coordinate grid is that one square unit on the grid is equivalent to one square
60
centimeter.
1.8.3 Identifying Your Strategy
At this point, you have simplified the problem by asking yourself questions about it, and created different
representations of the important information. The time has come to establish a formal plan of attack. This
is a crucial step in the problem-solving process, as it lays the groundwork for your solution.
To organize your thoughts, think of your geometric knowledge as a toolbox. Each time you learn a new
strategy, technique, or concept, add it to your toolbox. Then, when you need to solve a problem, you can
select the appropriate tool to use.
For now, take a quick look at the representations drawn for the example problem to identify what tools you
might need. You can use this section to clearly identify your strategy.
Example
Ehab drew a rectangle
in the diagonal
on the chalkboard.
was
and
was
. If Ehab draws
, what will be its length?
In the first representation, there is simply a rectangle with a diagonal. Though there is a way to solve this
problem using this diagram, it will not be covered until later in this book. For now, you do not have the tools
to solve it.
The second diagram shows the building blocks that comprise the rectangle. The diagonal cuts through the
blocks but presents the same challenges as the first diagram. You do not yet have the tools to solve the
problem using this diagram either.
The third diagram shows a coordinate grid with the rectangle drawn in. The diagonal has two endpoints
with specific coordinate pairs. In this chapter, you learned the distance formula to find lengths on a coordinate
grid. This is the tool you need to solve the problem.
Your strategy for this problem is to identify the two endpoints of
on the grid as
. Use the distance formula to find the length. The result will be the solution to the problem.
and
1.8.4 Making Calculations
The last step in any problem-solving situation is employing your strategy to find the answer. Be sure that
you use the correct values as identified in the relevant information. When you perform calculations, use a
pencil and paper to keep track of your work. Many careless mistakes result from mental calculations. Keep
track of each step along the way.
Finally, when you have found the answer, there are two more questions to ask yourself:
1. Did I provide the information the problem requested?
Go back to the first stages of the problem. Verify that you answered all parts of the question.
2. Does my answer make sense?
Your answer should make sense in the context of the problem. If your number is abnormally large or small
in value, check your work.
Example
61
Ehab drew a rectangle
in the diagonal
on the chalkboard.
was
and
was
. If Ehab draws
, what will be its length?
At this point, we have distilled the problem, created multiple representations of the scenario, and identified
the desired strategy. It is time to solve the problem.
The diagram below shows the rectangle on the coordinate grid.
To find the length of
, you must identify its endpoints on the grid. They are (1,1) and (9,7). Use the
, 1 for
, 9 for
, and 7 for
.
distance formula and substitute 1 for
is 10 cm.
Finally, make sure to ask yourself two more questions to verify your answer.
1. Did I provide the information the problem requested?
The problem asked you to identify the length of
. That is the information provided with our solution.
2. Does my answer make sense?
The value of 10 cm is slightly larger than 6 cm or 8 cm, but that is to be expected in this scenario. It is certainly
within reason. A response of 80 cm or 0.08 cm would have been unreasonable.
62
Your work on this problem is now complete. The final answer is 10 cm.
Lesson Summary
In this lesson, we explored problem-solving strategies. Specifically, we have learned:
•
How to read and understand given problem situations.
•
How to use multiple representations to restate problem situations.
•
How to identify problem-solving plans.
•
How to solve real-world problems using planning strategies.
These skills are important for any type of problem, whether or not it is about geometry. Practice breaking
down different problems in other parts of your life using these techniques. Forming plans and using strategies
will help you in a number of different ways.
Points to Consider
This chapter focused on the basic postulates of geometry and the most common vocabulary and notations
used throughout geometry. The following chapters focus on the skills of logic, reasoning, and proof. Review
the material in this chapter whenever necessary to maintain your understanding of the basic geometric
principles. They will be necessary as you continue in your studies.
1.8 Lesson Exercises
1. Suppose one line is drawn in a plane. How many regions of the plane are created?
2. Suppose two lines intersect in a plane. How many regions is the plane divided into? Draw a diagram of
your answer.
3. Now suppose three coplanar lines intersect at the same point in a plane. How many regions is the plane
divided into? Draw a diagram of your answer.
4. Make a table for the case of 4, 5, 6, and 7 coplanar lines intersecting at one point.
5. Generalize your answer for number 4. If
__________ regions.
coplanar lines intersect at one point, the plane is divided into
6. Bindi lives twelve miles south of Cindy. Mari lives five miles east of Bindi. What is the distance between
Cindy's house and Mari’s house?
a. Model this problem by drawing it on a coordinate grid. Let Bindi’s house bet at the origin, (0,0). Use the
labels
for Bindi’s house,
for Mari’s house, and
for Cindy’s house.
63
b. What are the coordinates of Cindy's and Mari’s house?
c. Use the distance formula to find the distance between
7. Suppose a camper is standing 100 meters north of a river that runs east-west in a perfectly straight line
(we have to make some assumptions for geometric modeling!). Her tent is 25 meters north of the river, but
300 meters downstream. See the diagram below).
The camper sees that her tent has caught fire! Luckily she is carrying a bucket so she can get water from
the river to douse the flames. The camper will run from her current position to the river, pick up a bucket of
water, and then run to her tent to douse the flames (see the blue line in the diagram). But how far along the
river should she run (distance
in the diagram) to pick up the bucket of water if she wants to minimize the
total distance she runs? Solve this by any means you see fit—use a scale model, the distance formula, or
some other geometric method.
8. Does it make sense for the camper in problem 7 to want to minimize the total distance she runs? Make
an argument for or against this assumption. (Note that in real-life problem solving finding the “best” answer
is not always simple!).
Answers
1. 2
2. 4
3. 6
64
4. See the table below
Number of Coplanar Lines Intersecting at One Point Number of Regions Plane is Divided Into
1
2
2
4
3
6
4
8
5
10
6
12
7
14
5. Every number in the right-hand column is two times the number in the left-hand column, so the general
coplanar lines intersect at one point, the plane is divided into
regions.”
statement is: “If
6.
a.
b. Cindy’s House: (0,12); Mari’s house: (5,0)
c. 13 miles
7. One way to solve this is to use a scale model and a ruler. Let 1 cm = 100 m. Then you can draw a picture
and measure the distance the camper has to run for various locations of the point where she gets water.
Be careful using the scale!
65
Now make a table for all measurements to find the best, shortest total distance.
x (meters)
0
25
50
100
125
150
175
200
225
250
275
300
Distance to Water (m)
100
103
112
141
160
180
202
224
246
269
293
316
Distance from Water to Tent (m)
301
276
251
202
177
152
127
103
79
56
35
25
Total Distance (m)
401
379
363
343
337
332
329
327
325
325
328
341
It looks like the best place to stop is between 225 and 250 meters. Based on other methods (which you will
learn in calculus and some you will learn later in geometry), we can prove that the best distance is when
she runs 240 meters downstream to pick up the bucket of water.
8. Answers will vary. One argument for why it is not best to minimize total distance is that she may run slower
with the full bucket of water, so she should take the distance she must run with a full bucket into account.
66
2. Reasoning and Proof
2.1 Inductive Reasoning
Learning Objectives
•
Recognize visual patterns and number patterns.
•
Extend and generalize patterns.
•
Write a counterexample to a pattern rule.
Introduction
You learned about some of the basic building blocks of geometry in Chapter 1. Some of these are points,
lines, planes, rays, and angles. In this section we will begin to study ways we can reason about these
building blocks.
One method of reasoning is called inductive reasoning. This means drawing conclusions based on examples.
2.1.1 Visual Patterns
Some people say that mathematics is the study of patterns. Let’s look at some visual patterns. These are
patterns made up of shapes.
Example 1
A dot pattern is shown below.
A. How many dots would there be in the bottom row of a fourth pattern?
. There is one more dot in the bottom row of each figure than in the previous figure. Also, the number of
dots in the bottom row is the same as the figure number.
B. What would the total number of dots be in the bottom row if there were 6 patterns?
. The rows would contain
The total number of dots is
and
dots.
.
Example 2
67
Next we have a pattern of squares and triangles.
A. How many triangles would be in a tenth illustration?
. There are
squares, with a triangle above and below each square. There is also a triangle on each
end of the figure. That makes
triangles in all.
B. One of the figures would contain 34 triangles. How many squares would be in that figure?
. Take off one triangle from each end. This leaves
triangles. Half of these
triangles, or
gles, are above and
triangles are below the squares. This means there are
squares.
To check: With
squares, there is a triangle above and below each square, making
Add one triangle for each end and we have
trian-
squares.
triangles in all.
C. How can we find the number of triangles if we know the figure number?
Let
be the figure number. This is also the number of squares.
below the squares. Add for the triangles on the ends.
If the figure number is
, then there are
is the number of triangles above and
triangles in all.
Example 3
Now look at a pattern of points and line segments.
For two points, there is one line segment with those points as endpoints.
For three noncollinear points (points that do not lie on a single line), there are three line segments with
those points as endpoints.
A. For four points, no three points being collinear, how many line segments with those points as endpoints
are there?
68
. The segments are shown below.
B. For five points, no three points being collinear, how many line segments with those points as endpoints
are there?
. When we add a th point, there is a new segment from that point to each of the other four points.
We can draw the four new dashed segments shown below. Together with the six segments for the four
points in part A, this makes
segments.
2.1.2 Number Patterns
You are already familiar with many number patterns. Here are a few examples.
Example 4 – Positive Even Numbers
The positive even numbers form the pattern
What is the
positive even number?
. Each positive even number is
times, to get the
Notice that the
more than the preceding one. You could start with
, then add
,
number. But there is an easier way, using more advanced mathematical thinking.
even number is
, the
even number is
, and so on. So the
even
number is
Example 5 – Odd Numbers
69
Odd numbers form the pattern
A. What is the
odd number?
. We can start with
number is
and add
times.
. Or, we notice that each odd
less than the corresponding even number. The
odd number is
4), so the
B. What is the
. The
even number is
(example
.
odd number?
even number is
(example 4), so the
odd number is
Example 6 – Square Numbers
Square numbers form the pattern
These are called square numbers because
A. What is the
. The
B. The
. The
square number?
square number is
square number is
=
.
. What is the value of
square number is
=
?
.
2.1.3 Conjectures and Counterexamples
A conjecture is an “educated guess” that is often based on examples in a pattern. Examples suggest a relationship, which can be stated as a possible rule, or conjecture, for the pattern.
Numerous examples may make you strongly believe the conjecture. However, no number of examples can
prove the conjecture. It is always possible that the next example would show that the conjecture does not
work.
Example 7
Here’s an algebraic equation.
Let’s evaluate this expression for some values of
These results can be put into a table.
70
.
After looking at the table, we might make this conjecture:
The value of
is
However, if we try other values of
for any whole number value of
, such as
.
, we have
Obviously, our conjecture is wrong. For this conjecture,
is called a counterexample, meaning that
this value makes the conjecture false. (Of course, it was a pretty poor conjecture to begin with!)
Example 8
Ramona studied positive even numbers. She broke some positive even numbers down as follows:
What conjecture might be suggested by Ramona’s results?
Ramona made this conjecture:
“Every positive even number is the sum of two different positive odd numbers.”
Is Ramona’s conjecture correct? Can you find a counterexample to the conjecture?
The conjecture is not correct. A counterexample is
that is equal to
is:
. The only way to make a sum of two odd numbers
, which is not the sum of different odd numbers.
Example 9
Artur is making figures for a graphic art project. He drew polygons and some of their diagonals.
Based on these examples, Artur made this conjecture:
If a convex polygon has
sides, then there are
diagonals from any given vertex of the polygon.
Is Artur’s conjecture correct? Can you find a counterexample to the conjecture?
The conjecture appears to be correct. If Artur draws other polygons, in every case he will be able to draw
diagonals if the polygon has
sides.
71
Notice that we have not proved Artur’s conjecture. Many examples have (almost) convinced us that it is
true.
Lesson Summary
In this lesson you worked with visual and number patterns. You extended patterns to beyond the given items
and used rules for patterns. You also learned to make conjectures and to test them by looking for counterexamples, which is how inductive reasoning works.
Points to Consider
Inductive reasoning about patterns is a natural way to study new material. But we saw that there is a serious
limitation to inductive reasoning: No matter how many examples we have, examples alone do not prove
anything. To prove relationships, we will learn to use deductive reasoning, also known as logic.
2.1 Lesson Exercises
How many dots would there be in the fourth pattern of each figure below?
1.
2.
3.
4. What is the next number in the following number pattern?
5. What is the tenth number in this number pattern?
The table below shows a number pattern.
6. What is the value of
72
when
?
7. What is the value of
8. Is
a value of
when
?
in this pattern? Explain your answer.
Give a counterexample for each of the following statements.
9. If
is a whole number, then
10. Every prime number is an odd number.
11. If
and
, then
.
Answers
1.
2.
3.
4.
5.
6.
7.
8. No. Values of
are
that makes
value of
9.
10.
or,
there is no
.
because
because
is prime but not odd.
11. Any set of points where
and
are not collinear.
2.2 Conditional Statements
Learning Objectives
•
Recognize if-then statements.
•
Identify the hypothesis and conclusion of an if-then statement.
•
Write the converse, inverse, and contrapositive of an if-then statement.
73
•
Understand a biconditional statement.
Introduction
In geometry we reason from known facts and relationships to create new ones. You saw earlier that inductive
reasoning can help, but it does not prove anything. For that we need another kind of reasoning. Now you
will begin to learn about deductive reasoning, the kind of reasoning used throughout mathematics and
science.
2.2.1 If-Then Statements
In geometry, and in ordinary life, we often make conditional, or if-then, statements.
•
Statement 1: If the weather is nice, I’ll wash the car. (“Then” is implied even if not stated.)
•
Statement 2: If you work overtime, then you’ll be paid time-and-a-half.
•
Statement 3: If
•
Statement 4: If a triangle has three congruent sides, it is an equilateral triangle. (“Then” is implied; this
is a definition.)
•
Statement 5: All equiangular triangles are equilateral. (“If” and “then” are both implied.)
divides evenly into
, then
is an even number.
An if-then statement has two parts.
•
The “if” part is called the hypothesis.
•
The “then” part is called the conclusion.
For example, in statement 2 above, the hypothesis is “you work overtime.” The conclusion is “you’ll be paid
time-and-a-half.”
Look at statement 1 above. Even though the word “then” is not actually present, the statement could be
rewritten as: If the weather is nice, then I’ll wash the car. This is the meaning of statement 1. The hypothesis
is “the weather is nice.” The conclusion is “I’ll wash the car.”
Statement 5 is a little more complicated. “If” and “then” are both implied without being stated. Statement 5
can be rewritten as: If a triangle is equiangular, then it is equilateral.
What is meant by an if-then statement? Suppose your friend makes the statement in statement 2 above,
and adds another fact.
•
If you work overtime, then you’ll be paid time-and-a-half.
•
You worked overtime this week.
If we accept these statements, what other fact must be true? Combining these two statements, we can state
with no doubt:
You’ll be paid time-and-a-half this week.
Let’s analyze statement 1, which was rewritten as: If the weather is nice, then I’ll wash the car. Suppose we
accept statement 1 and another fact: I’ll wash the car.
Can we conclude anything further from these two statements? No. Even if the weather is not nice, I might
wash the car. We do know that if the weather is nice I’ll wash the car. We don’t know whether or not I might
74
wash the car even if the weather is not nice.
2.2.2 Converse, Inverse, and Contrapositive of an If-Then
Statement Look at statement 1 above again.
If the weather is nice, then I’ll wash the car.
This can be represented in a diagram as:
If
then
.
= the weather is nice
“If
then
= I’ll wash the car
” is also written as
Notice that conditional statements, hypotheses, and conclusions may be true or false.
then may be true or false.
“If
and the statement
In deductive reasoning we sometimes study statements related to a given if-then statement. These are
and their opposites, or negations (“not”). Note that “not ” is written in symbols as
formed by using
.
and
can be combined to produce new if-then statements.
•
The converse of
•
The inverse of
•
The contrapositive of
is
is
is
Now let’s go back to statement 1: If the weather is nice, then I’ll wash the car.
Converse:
=
the weather is nice
=
I’ll wash the car
=
the weather is not nice
=
I’ll wash the car (or I wash the car)
=
I won’t wash the car (or I don’t wash the car)
If I wash the car, then the weather was nice.
Inverse:
If the weather is not nice, then I won’t wash the car.
Contrapositive:
If I don’t wash the car, then the weather is not nice.
Notice that if we accept statement 1 as true, then the converse and inverse may, or may not, be true. But
the contrapositive is true. Another way to say this is: The contrapositive is logically equivalent to the original
if-then statement. In future work you may be asked to prove an if-then statement. If it’s easier to prove the
75
contrapositive, then you can do this since the statement and its contrapositive are equivalent.
Example 1
Statement:
Converse:
If
then
. True.
, then
If
A counterexample is
. False.
, where
Inverse:
If
is not
then
A counterexample is
where
Contrapositive:
but
is not
. False.
is not
If
is not
, then
If
is not
, then
is not
but
is not
=
. True.
and
is not
Example 2
Statement:
then
If
is the midpoint of
. False (as shown below).
Needs
Converse:
Inverse:
Contrapositive:
If
is the midpoint of
If
If
, then
, then
. True.
is not the midpoint of
is not the midpoint of
, then
. True.
False (see the diagram above).
2.2.3 Biconditional Statements
You recall that the converse of “If
then
” is “If
then
biconditional statement.
Biconditional:
In symbols, this is written as:
76
and
.” When these two are combined, we have a
We read
as:
“
if and only if
”
Example 3
True statement:
if and only if
is an obtuse angle.
You can break this down to say:
If
then
is an obtuse angle and if
is an obtuse angle then
.
Notice that both parts of this biconditional are true; the biconditional itself is true.
You most likely recognize this as the definition of an obtuse angle.
Geometric definitions are biconditional statements that are true.
Example 4
Let
be
Let
be
a. Is
true?
Yes.
is if
then
From algebra we know that if
So if
then
b. Is
true?
then
, or
and
If
, then we know that
is true.
No.
is if
, then
From algebra we know that if
However,
, then
does not guarantee that
can be less that
So if
but still not less than
, then
, or
, for example if
is
.
, is false.
true?
c. Is
No.
is
if and only if
77
We saw above that the if part of this statement, which is
If
then
This statement is false. One counterexample is
Note that if either
or
.
is false, then
is false.
Lesson Summary
In this lesson you have learned how to express mathematical and other statements in if-then form. You also
learned that each if-then statement is linked to variations on the basic theme of “If then .” These variations
are the converse, inverse, and contrapositive of the if-then statement. Biconditional statements combine the
statement and its converse into a single “if and only if” statement. Definitions are an important type of biconditional, or if-and-only-if, statement.
Points to Consider
We called points, lines, and planes the building blocks of geometry. We will soon see that hypothesis, conclusion, as well as if-then and if-and-only-if statements are the building blocks that deductive reasoning, or
logic, is built on. This type of reasoning will be used throughout your study of geometry. In fact, once you
understand logical reasoning you will find that you apply it to other studies and to information you encounter
all your life.
2.2 Lesson Exercises
Write the hypothesis and the conclusion for each statement.
1. If
divides evenly into
, then
is an even number.
2. If a triangle has three congruent sides, it is an equilateral triangle.
3. All equiangular triangles are equilateral.
4. What is the converse of the statement in exercise 1 above? Is the converse true?
5. What is the inverse of the statement in exercise 2 above? Is the inverse true?
6. What is the contrapositive of the statement in exercise 3? Is the contrapositive true?
7. The converse of a statement about collinear points
is the midpoint of
•
What is the statement?
•
Is it true?
,
, and
is: If
and
.
8. What is the inverse of the inverse of if
then
?
9. What is the one-word name for the converse of the inverse of an if-then statement?
10. What is the one-word name for the inverse of the converse of an if-then statement?
For each of the following biconditional statements:
78
, then
Write
in words.
Write
in words.
Is
true?
Is
true?
Is
true?
Note that in these questions,
and
could be reversed and the answers would be correct.
11. A U.S. citizen can vote if and only if he or she is
or more years old.
12. A whole number is prime if and only if it is an odd number.
13. Points are collinear if and only if there is a line that contains the points.
14.
if only if
and
Answers
1. Hypothesis:
divides evenly into
; conclusion:
is an even number.
2. Hypothesis: A triangle has three congruent sides; conclusion: it is an equilateral triangle.
3. Hypothesis: A triangle is equiangular; conclusion: the triangle is equilateral.
4. If
is an even number, then
divides evenly into
. True.
5. If a triangle does not have three congruent sides, then it is not an equilateral triangle. True.
6. If a triangle is not equilateral, then it is not equiangular. True.
7. If
is the midpoint of
, etc.).
8. If
, then
and
. False (
and
could both be
,
then
9. Contrapositive
10. Contrapositive
11.
he or she is
is true.
12.
a whole number is an odd number;
is false.
13.
is true.
14.
or more years old;
a line contains the points;
and
;
a U. S. citizen can vote;
is true;
is true;
a whole number is prime;
is false;
is false;
the points are collinear; is
;
is true;
is true;
is false;
is true;
is false.
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2.3 Deductive Reasoning
Learning Objectives
•
Recognize and apply some basic rules of logic.
•
Understand the different parts that inductive reasoning and deductive reasoning play in logical reasoning.
•
Use truth tables to analyze patterns of reasoning.
Introduction
You began to study deductive reasoning, or logic, in the last section, when you learned about if-then statements. Now we will see that logic, like other fields of knowledge, has its own rules. When we follow those
rules, we will expand our base of facts and relationships about points, lines, and planes. We will learn two
of the most useful rules of logic in this section.
2.3.1 Direct Reasoning
We all use logic—whether we call it that or not—in our daily lives. And as adults we use logic in our work
as well as in making the many decisions a person makes every day.
•
Which product should you buy?
•
Who should you vote for?
•
Will this steel beam support the weight you place on it?
•
What will be your company’s profit next year?
Let’s see how common sense leads to the two most basic rules of logic.
Example 1
Suppose Bea makes the following statements, which are known to be true.
If Central High School wins today, they will go to the regional tournament.
Central High School does win today.
Common sense tells us that there is an obvious logical conclusion if these two statement are true:
Central High School will go to the regional tournament.
Example 2
Here are two true statements.
is an odd number.
Every odd number is the sum of an even and an odd number.
Based on only these two true statements, there is an obvious further conclusion:
is the sum of an even and an odd number.
(this is true, since
80
).
Example 3
Suppose the following two statements are true.
1. If you love me let me know, if you don’t then let me go. (A country music classic. Lyrics by John Rostill.)
2. You don’t love me.
What is the logical conclusion?
Let me go.
There are two statements in the first line. The second one is:
If you don’t (love me) then let me go.
You don’t love me is stated to be true in the second line.
Based on these true statements, Let me go is the logical conclusion.
Now let’s look at the structure of all of these examples, using the
and
symbols that we used earlier.
Each of the examples has the same form.
conclusion :
A more compact form of this argument, (logical pattern) is:
To state this differently, we could say that the true statement
and .
follows automatically from the true statements
This reasoning pattern is one of the basic rules of logic. It’s called the law of detachment.
Law of Detachment Suppose and
and You can conclude
are statements. Then given
Practice saying the law of detachment like this: “If
is true, and
is true, then
is true.”
Example 4
81
Here are two true statements.
If
and
and
are a linear pair, then
.
are a linear pair.
What conclusion do we draw from these two statements?
The next example is a warning not to turn the law of detachment around.
Example 5
Here are two true statements.
If
and
are a linear pair, then
and
What conclusion can we draw from these two statements?
None! These statements are in the form
Note that since
and
does not mean that they are a linear pair.
, we also know that
, but this
The law of detachment does not apply. No further conclusion is justified.
You might be tempted to conclude that
and
are a linear pair, but if you think about it you will realize that would not be justified. For example, in the rectangle below
and
(and
, but
, and
are definitely NOT a linear pair.
Now let’s look ahead. We will be doing some more complex deductive reasoning as we move ahead in geometry. In many cases we will build chains of connected if-then statements, leading to a desired conclusion.
Start with a simplified example.
Example 6
Suppose the following statements are true.
1. If Pete is late, Mark will be late.
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2. If Mark is late, Wen will be late.
3. If Wen is late, Karl will be late.
To these, add one more true statement.
4. Pete is late.
One clear consequence is: Mark will be late. But make sure you can see that Wen and Karl will also be
late.
Here’s a symbolic form of the statements.
1.
2.
3.
4.
Our statements form a “chain reaction.” Each “then” becomes the next “if” in a chain of statements. The
chain can consist of any number of connected statements. Once we add the true
statement as above,
we know that the conclusion (the then part) of the last statement is justified.
Another way to look at this is to imagine a chain of dominoes. The dominoes are the linked if-then statements.
Once the first domino falls, each domino knocks the next one over, and the last domino falls. is the tipping
over of the first domino. The final conclusion of the last if-then statement is the last domino.
This is called the law of syllogism. A formal statement of this rule of logic is given below.
Law of Syllogism Suppose
,
, ...,
, and
are statements. Then given that
is true and that you have the following relationship:
Then, you can conclude
2.3.2 Inductive vs. Deductive Reasoning
You have now worked with both inductive and deductive reasoning. They are different but not opposites. In
fact, they will work together as we study geometry and other mathematics.
How do these two kinds of reasoning complement (strengthen) each other? Think about the examples you
saw earlier in this chapter.
83
Inductive reasoning means reasoning from examples. You may look at a few examples, or many. Enough
examples might make you suspect that a relationship is true always, or might even make you sure of this.
But until you go beyond the inductive stage, you can’t be absolutely sure that it is always true.
That’s where deductive reasoning enters and takes over. We have a suggestion arrived at inductively. We
then apply rules of logic to prove, beyond any doubt, that the relationship is true always. We will use the law
of detachment and the law of syllogism, and other logic rules, to build these proofs.
2.3.3 Symbolic Notation and Truth Tables
Logic has its own rules and symbols. We have already used letters like and to represent statements:
for the negation (“not”), and the arrow
to indicate if-then. Here are two more symbols we can use.
= and
= or
Truth tables are a way to analyze statements in logic. Let’s look at a few simple truth tables.
Example 1
How is
related to logically? We make a truth table to find out. Begin with all the possible truth values
of . This is very simple; can be either true (T), or false (F).
T
F
Next we write the corresponding truth values for
has the opposite truth value as
then
is false, and vice versa. Complete the truth table by filling in the
column.
T
F
F
T
Now we construct truth tables for slightly more complex logic.
Example 2
Draw a truth table for
and
written
.
Begin by filling in all the T/F combinations possible for
84
T
T
T
F
F
T
F
F
and
.
. If
is true,
How can and be true? Common sense tells us that
We complete the last column accordingly.
T
T
T
T
F
F
F
T
F
F
F
F
and
Another way to state the meaning of the truth table is that
is false whenever either
is true only when
or
is true and
is false.
is true.
Let’s do the same for
or
. Before we do that, we need to clarify which “or” we mean in mathematics.
In ordinary speech, or is sometimes used to mean, “this or that, but not both.” This is called the exclusive
or (it excludes or keeps out both). In mathematics, or means “this, that, or both this and that.” This is called
the inclusive or. Knowing that or is inclusive makes the truth table an easy job.
Example 2
or
is true, because
or
is true, because
or
is true because
or
is false because
is true.
is true.
is true and
is false and
is true.
is false.
Example 3
Draw a truth table for
or
, which is written
.
Begin by filling in all the T/F combinations possible for and . Keeping in mind the definition of or above
(inclusive), fill in the third column. or will only be false when both and are false; it is true otherwise.
T
T
T
T
F
T
F
T
T
F
F
F
Lesson Summary
Do we all have our own version of what is logical? Let’s hope not—we wouldn’t be able to agree on what is
or isn’t logical! To avoid this, there are agreed-on rules for logic, just like there are rules for games. The two
most basic rules of logic that we will be using throughout our studies are the law of detachment and the
law of syllogism.
Points to Consider
Rules of logic are universal; they apply to all fields of knowledge. For us, the rules give a powerful method
for proving new facts that are suggested by our explorations of points, lines, planes, and so on. We will
structure a specific format, the two-column proof, for proving these new facts. In upcoming lessons you will
85
write two-column proofs. The facts or relationships that we prove are called theorems.
2.3 Lesson Exercises
Must the third sentence be true if the first two sentences are true? Explain your answer.
1. People who vote for Jane Wannabe are smart people.
I am a smart person.
I will vote for Jane Wannabe.
2. If Rae is the driver today then Maria is the driver tomorrow.
Ann is the driver today.
Maria is not the driver tomorrow.
3. All equiangular triangles are equilateral.
is equiangular.
is equilateral.
What additional statement must be true if the given sentences are true?
4. If West wins, then East loses.
If North wins, then West wins.
5. If
then
If
then
.
Fill in the truth tables.
6.
T
F
7.
T
F
8.
86
.
.
T
T
T
F
F
T
F
F
9.
T
T
T
F
F
T
F
F
10. When is
true?
11. For what values of
is the following statement true?
or
12. For what values of
is the following statement true?
or
Answers
1. No (converse error).
2. No (inverse error).
3. Yes.
4. If North wins, then East loses.
5.
. (also
)
6.
T
F
F
F
T
F
Note that
is never true.
7.
87
T
F
T
F
T
T
Note that
is always true.
8.
T
T
F
F
F
T
F
F
T
F
F
T
T
F
F
F
F
T
T
T
Note that
is true only when
and
are both false.
9.
T
T
F
T
T
T
F
T
T
T
F
T
F
T
F
F
F
T
T
F
10.
is always true except when
,
, and
are all false.
11.
12. none,
2.4 Algebraic Properties
Learning Objectives
•
Identify and apply properties of equality.
•
Recognize properties of congruence “inherited” from the properties of equality.
•
Solve equations and cite properties that justify the steps in the solution.
•
Solve problems using properties of equality and congruence.
Introduction
We have begun to assemble a toolbox of building blocks of geometry (points, lines, planes) and rules of
logic that govern deductive reasoning. Now we start to expand our geometric knowledge by applying logic
to the geometric building blocks. We’ll make a smooth transition as some fundamental principles of algebra
88
take on new life when expressed in the context of geometry.
2.4.1 Properties of Equality
All things being equal, in mathematics the word “equal” means “the same as.” To be precise, the equal sign
means that the expression on the left of the equal sign and the expression on the right represent the
same number. So equality is specifically about numbers—numbers that may be expressed differently but
are in fact the same.
Some examples:
•
•
•
Basic properties of equality are quite simple and you are probably familiar with them already. They are listed
here in formal language and then translated to common sense terms.
Properties of Equality
For all real numbers
•
,
, and
:
Reflexive Property:
That is, any number is equal to itself, or the same as itself.
Example:
•
Symmetric Property: If
then
.
You can read an equality left to right, or right to left.
Example: If
then
Example: If
, then
Sometimes it is more convenient to write
•
Transitive Property: If
and
.
than
then
. The symmetric property allows this.
.
Translation: If there is a “chain” of linked equations, then the first number is equal to the last number. (You
can prove that this applies to more than two equalities in the review questions.)
Example: If
and
, then
.
As a reminder, here are some properties of equality that you used heavily when you learned to solve
equations in algebra.
•
Substitution Property: If
Example: Given that
then
and that
can be put in place of
. Then
anywhere or everywhere.
.
89
•
Addition Property of Equality: If
, then
.
Translation: You can add the same number to both sides of an equation.
Example: If
•
, then
.
Multiplication Property of Equality: If
, then
Translation: You can multiply the same number on both sides of an equation.
Example: If
, then
.
Keep in mind that these are properties about numbers. As you go further into geometry, you can apply the
properties of equality to anything that is a number: lengths of segments and angle measures, for example.
2.4.2 Properties of Congruence
Let’s review the definitions of congruent segments and angles.
Congruent Segments:
if and only if
Remember that, although
.
and
of those segments, meaning that
equality apply to
Congruent Angles:
and
if and only if
are segments,
and
and
are lengths
are numbers. The properties of
.
=
The comment above about segment lengths also applies to angle measures. The
properties of equality apply to
and
.
Any statement about congruent segments or congruent angles can be translated directly into a statement
about numbers. This means that each property of equality has a corresponding property of congruent segments and a corresponding property of congruent angles.
Here are some of the basic properties of equality and the corresponding congruence properties.
Given that
and
are real numbers.
Reflexive Property of Equality:
Reflexive Property of Congruence of Segments:
Reflexive Property of Congruence of Angles:
Symmetric Property of Equality: If
, then
Symmetric Property of Congruence of Segments:
90
.
If
, then
Symmetric Property of Congruence of Angles:
If
and
Transitive Property of Equality: If
, then
.
, then
Transitive Property of Congruence of Segments
If
and
then
Transitive Property of Congruence of Angles
If
, and
then
.
2.4.3 Using Congruence Properties in Equations
When you solve equations in algebra you use properties of equality. You might not write out the logical
justification for each step in your solution, but you know that there is an equality property that justifies that
step.
Let’s see how we can use the properties of congruence to justify statements in deductive reasoning. Abbreviated names of the properties can be used.
Example 1
Given points
Are
,
, and
, and
, with
, and
.
collinear?
(reflexive).
Why do we want this? So that we can bring
in the numbers that are
and
.
Justification is substitution of
for
.
for
and
; this is arithmetic. No justification is needed as long as the arithmetic
is correct.
More arithmetic.
Substituting
for
.
, and
are not collinear.
for
Segment addition postulate.
are collinear if and
and
, and
only if
.
Example 2
Given that
Prove that
and
.
is an acute angle.
91
These are the given facts.
,
Substitute
property.
for
using the transitive
Addition property of equality; add
both sides.
to
Arithmetic.
More arithmetic.
Substitute
for
.
Definition. An angle is acute if and only if its
measure is between
and
.
is an acute angle.
The deductive reasoning scheme in example 2 is called a proof. The final statement must be true if the
given information is true.
Lesson Summary
We built on our previous knowledge of properties of equality to derive corresponding properties of congruence.
This enabled us to test statements about congruence, and to create new properties and relationships about
congruence. We had our first introduction, in informal terms, to logical proof.
Points to Consider
In the examples and review questions, terms like given, prove, and reason were used. In upcoming lessons
we’ll see how to identify the given facts, how to draw a diagram to represent a statement that we need to
prove, and how to organize proofs more formally. As we move ahead we’ll prove many important geometric
relationships called theorems. We have now laid the framework of logic that we’ll use repeatedly in future
work.
2.4 Lesson Exercises
Given:
, and
are real numbers.
Use the given property or properties of equality to fill in the blank in each of the following questions.
1. Symmetric: If
, then ___________________.
, then ___________________.
2. Reflexive: If
3. Transitive: If
and
then __________________.
4. Symmetric: If
, then ____________________.
5. Reflexive: If
, then _____________________.
6. Substitution: If
and
, then _________________________.
7. Use the transitive property of equality to write a convincing logical argument (a proof) that the statement
below is true.
If
92
and
and
and
, then
.
Note that this chain could be extended with additional links.
Let
be the relation “is the mother of.” Let
8.Is
symmetric? Explain your answer.
9. Is
symmetric? Explain your answer.
10. Is
transitive? Explain your answer.
11. Is
transitive? Explain your answer.
and
12. Let
be the relation “is the brother of.”
be real numbers. Prove: If
and
, then
13. The following statement is not true. “Let
and
be points. If
and
, then
” Draw a diagram with these points shown to provide a counterexample.
Answers
1.
2.
3.
4.
5.
6.
( or
and
7. If
property). If
).
then
and
(transitive property). If
then
(transitive property).
and
then
(transitive
8. No. If Maria is the mother of Juan, it does NOT follow that Juan is the mother of Maria!
9. Yes. For example, if Bill is Frank’s brother, then Frank is Bill’s brother.
10. No. If
were transitive, then “Maria is Fern’s mother and Fern is Gina’s mother” would lead to “Maria
is Gina’s mother.” However, Maria would actually have to be Gina’s grandmother!
11. Yes. If Bill is Frank’s brother and Frank is Greg’s brother, the Bill is Greg’s brother. You might say the
brother of my brother is (also) my brother.
12.
for
and
(given);
(reflexive);
(substitute
for
and
).
13. Below is an example:
93
A correct response is a diagram showing:
•
.
•
.
•
If
.
,
, and
are collinear and
,
, and
are not collinear then the conditions are satisfied.
2.5 Diagrams
Learning Objectives
•
Provide the diagram that goes with a problem or proof.
•
Interpret a given diagram.
•
Recognize what can be assumed from a diagram and what can not be.
•
Use standard marks for segments and angles in diagrams.
Introduction
Geometry is about objects such as points, lines, segments, rays, planes, and angles. If we are to solve
problems about these objects, our work is made much easier if we can represent these objects in diagrams.
In fact, for most of us, diagrams are absolutely essential for problem solving in geometry.
2.5.1 Basic Postulates—Another Look
Just as undefined terms are building blocks that other definitions are built on, postulates are the building
blocks of logic. We’re now ready to restate some of the basic postulates in slightly more formal terms, and
to use diagrams.
Postulate 1
Through any two distinct points, there is exactly one line.
Comment: Any two points are collinear.
Postulate 2
There is exactly one plane that contains any three noncollinear points.
Comment: Sometimes this is expressed as: “Three noncollinear points determine a
plane.”
94
Postulate 3
Postulate 4
If two points are in a plane, then the whole line through those two points is in the
plane.
If two distinct lines intersect, then the intersection is exactly one point.
Comments: Some lines intersect, some do not. If lines do intersect, it is in only one
point, otherwise one or both “lines” would have to curve, which lines do not do.
Postulate 5
If two distinct planes intersect, then the intersection is exactly one line.
Comments: Some planes intersect, some do not. Think of a floor and a ceiling as
models for planes that do not intersect. If planes do intersect, it is in a line. Think of
the edge of a box (a line) formed where two sides of the box (planes) meet.
Postulate 6
The Ruler Postulate: The points on a line can be assigned real numbers, so that for
any two points, one corresponds to 0 and the other corresponds to a nonzero real
number.
Comments: This is how a number line and a ruler work. This also means we can
measure any segment.
Postulate 7
and
are collinear if and only if
Comment: If
and
are not collinear, then
examples of this fact in earlier sections of this chapter.
. We saw
The Segment Addition Postulate: Points
.
Postulate 8
The Protractor Postulate: If rays in a plane have a common endpoint, can be
assigned to one ray and a number between and
can be assigned to each of
the other rays.
Comment: This means that any angle has a (degree) measure.
Postulate 9
The Angle Addition Postulate: Let
the interior of
and
if and only if
+
be points in a plane.
=
is in
.
Comment: If an angle is made up of other angles, the measures of the component
angles can be added to get the measure of the “big” angle.
Postulate 10
The Midpoint Postulate: Every line segment has exactly one midpoint.
Comments: If
is a point on
, let’s say point
Postulate 11
,with
and
there is not another point on
. The midpoint of a segment is unique.
The Angle Bisector Postulate: Every angle has exactly one bisector.
95
Comments: The bisector of an angle is a ray. If
bisects ,
there is not
another ray that bisects the angle. The bisector (ray) of an angle is unique.
2.5.2 Using Diagrams
Now we apply our definitions and postulates to a geometric figure. When measures are given on a figure,
we can assume that the measurements on the figure are correct. We can also assume that:
•
Points that appear to be collinear are collinear.
•
Lines, rays, or segments that appear to intersect do intersect.
•
A ray that appears to be in the interior of an angle is in the interior of the angle.
We cannot assume the following from a diagram:
•
That lines, segments, rays, or planes are parallel or perpendicular.
•
That segments or angles are congruent.
These must be stated or indicated in the diagram.
The diagram below shows some segment and angle measures.
Example 1
A. Is
No.
the midpoint of
is on
Explain your answer.
, but
.
B. Is
the midpoint of
Explain your answer.
Yes.
is on
=
, and
.
C. Name an angle bisector and the angle that it bisects.
bisects
96
.
D. Fill in the blank:
E. Is
No. If
=
+
the bisector of
bisected
_______.
Explain your answer.
, then
would be
That would make
but
Sometimes we use special marks in diagrams. Tick marks show congruent segments. Arc marks show
congruent angles. Right angle marks show right angles and perpendicular lines and segments.
When these signs are used, the relationships they represent become part of the given information for a
problem.
Example 2
Blue blobs are dots for points and single and double arc marks to show equal angles.
Based on the marks on the diagram, we know that:
(single tick marks).
(double tick marks).
=
(single arc marks).
=
(double arc marks).
Lesson Summary
As we move forward toward more formal reasoning, we have reviewed the basic postulates and expressed
them more formally. We saw that most geometric situations involve diagrams. In diagrams we can assume
some facts, and we cannot assume others.
Points to Consider
In upcoming lessons you will organize your reasoning pattern into the two-column proof. This is a traditional
pattern that still works very well today. It gives us a clear, direct format, and uses the basic rules of logic
that we saw in earlier lessons. We will prove many important geometric relationships called theorems
97
throughout the rest of this geometry course.
2.5 Lesson Exercises
Use the diagram to answer questions 1-8.
1. Name a right angle.
2. Name two perpendicular lines (not segments).
3. Given that
is
true? Explain your answer completely.
4. Given that
is
a rectangle? Explain your answer informally. (Note: This is a new
question. Do not assume that the given from a previous question is included in this question.)
5. Fill in the blanks:
. Why?
. Why?
6. Fill in the blanks:
7. Given that
8. Given that
, prove
, prove:
What geometric objects does the real-world model suggest?
9. Model: two railroad tracks
10. Model: a floor and a ceiling
11. Model: two lines on a piece of graph paper
12. Model: referee’s arms when signaling a touchdown
13. Model: capital letter
98
14. Model: the spine of a book where the front and back covers join
Answers
1.
2.
and
3. Yes
Given
Reflexive
Substitution
Segment addition postulate
Substitution
4. Yes. It’s given that
(so
must be equal to
). Since
, and this would make
and
then
a rectangle.
5.
6.
7.
Given
Given
Angle Addition Postulate
Angle Addition Postulate
Substitution
Substitution
Definition of congruent angles
8.
99
Given
Reflexive
Substitution
Segment addition postulate
Substitution
9. Parallel lines
10. Parallel planes
11. Parallel or perpendicular lines
12. Parallel lines or segments
13. Perpendicular segments
14. Intersecting planes
2.6. Two-Column Proof
Learning Objectives
•
Draw a diagram to help set up a two-column proof.
•
Identify the given information and statement to be proved in a two-column proof.
•
Write a two-column proof.
Introduction
You have done some informal proofs in earlier sections. Now we raise the level of formality higher. In this
section you will learn to write formal two-column proofs. You’ll need to draw a diagram, identify the given
and prove, and write a logical chain of statements. Each statement will have a reason, such as a definition,
postulate, or previously proven theorem, that justifies it.
2.6.1 Given, Prove, and Diagram
Example 1
Write a two-column proof for the following:
If
,
,
, and
are points on a line, in the given order, and
, then
.
Comments: The if part of the statement contains the given. The then part is the section that you must prove.
A diagram should show the given facts.
We start with the given, prove, and a diagram.
•
•
100
Given:
Prove:
, and
.
are points on a line in the order given.
.
points on the line;
Now it’s time to start with the given. Then we use logical reasoning to reach the statement we want to prove.
Often (not always) the proof starts with the given information.
In the two column format, Statements go on the left side, and Reasons for each statement on the right.
Reasons are generally definitions, postulates, and previously proved statements (called theorems).
Statement
Reason
1.
Given
2.
order
, and
are collinear in that
Given
Reflexive
3.
a n d Segment Addition Postulate
4.
5.
Addition Property of Equality
6.
Substitution
is what we were given to prove, and we’ve done it.
Example 2
Write a two-column proof of the following:
•
Given:
•
bisects
;
Prove:
Statement
Reason
Given
1.
bisects
2.
Definition of angle bisector
3.
Angle Addition Postulate
101
4.
5.
6.
7.
Angle Addition Postulate
Substitution
Given
Substitution
8.
Subtract
from both sides (Reminder: Angle measures are all real numbers, so properties of quality apply.)
9.
Definition of congruent angles
This is the end of the proof. The last statement is the requirement made in the prove above. This is the
signal that the proof is completed.
Lesson Summary
In this section you have seen two examples illustrating the format of two-column proofs. The format of twocolumn proofs is the same regardless of the specific details. Geometry originated many centuries ago using
this same kind of deductive reasoning proof.
Points to Consider
You will see and write many two-column proofs in future lessons. The framework will stay the same, but the
details will be different. Some of the statements that we prove are important enough that they are identified
by the name theorem. You will learn about many theorems and use them in proofs and problem solving.
2.6 Lesson Exercises
Use the diagram below to answer questions 1-10.
Which of the following can be assumed to be true from the diagram? Answer yes or no.
1.
2.
3.
4.
5.
102
6.
bisects
7.
8.
9.
is a square
10.
is a rectangle
Use the diagram below to answer questions 11-14.
Given:
bisects
,
is the midpoint of
, and
.
11. How many segments have two of the given points as endpoints?
What is the value of each of the following?
12.
13.
14.
15. Write a two-column proof for the following:
Given:
bisects
Prove:
Answers
1. No
2. No
3. Yes
4. No
103
5. Yes
6. No
7. No
8. No
9. No
10. No
11.
12.
13.
14.
15.
Statement
Reason
Given
1.
bisects
2.
=
3.
+
Definition of angle bisector
=
Angle Addition Postulate
4.
Given
5.
Definition of perpendicular segments
6.
+
Substitution
7.
Algebra (Distributive Property)
8.
Multiplication Property of Equality
2.7 Segment and Angle Congruence
Theorems Learning Objectives
•
Understand basic congruence properties.
•
Prove theorems about congruence.
Introduction
In an earlier lesson you reviewed many of the basic properties of equality. Properties of equality are about
numbers. Angles and segments are not numbers, but their measures are numbers. Congruence of angles
and segments is defined in terms of these numbers. To prove congruence properties, we immediately turn
104
congruence statements into number statements, and use the properties of equality.
2.7.1 Equality Properties
Reminder: Here are some of the basic properties of equality. These are postulates—no proof needed. For
each of these there is a corresponding property of congruence for segments, and one for angles. These are
theorems—we’ll prove them.
Properties of Equality for real numbers
•
Reflexive
•
Symmetric
If
then
•
Transitive
If
and
, and
.
, then
These properties are convertibles; we can convert them quickly and easily into congruence theorems.
Note that diagrams are needed to prove the congruence theorems. They are about angles and segments...ALL
angles and segments, wherever and whenever they are found. No special setting (diagram) is needed.
2.7.2 Segment Congruence Properties
In this section we’ll prove a series of segment theorems.
Reflexive:
Statement
Reason
1.
Reflexive Property of Equality
2.
Definition of congruent segments
Symmetric: If
, then
Given:
Prove:
Statement
Reason
1.
Given
2.
Definition of congruent segments
3.
Symmetric Property of Equality
4.
Definition of congruent segments
Transitive: If
and
, then
105
Given:
;
Prove:
Statement
Reason
1.
Given
;
Definition of congruent segments
2.
3.
Transitive property of equality
4.
Definition of congruent segments
2.7.3 Angle Congruence Properties
Watch for proofs of the Angle Congruence Properties in the Lesson Exercises.
Reflexive:
Symmetric:
If
, then
Transitive:
If
and
, then
Lesson Summary
In this lesson we looked at old information in a new light. We saw that the properties of equality—reflexive,
symmetric, transitive—convert easily into theorems about congruent segments and angles. In the next
section we’ll move ahead into new ground. There we’ll get to use all the tools in our geometry toolbox to
solve problems and to create new theorems.
Points to Consider
We are about to transition from introductory concepts that are necessary but not too “geometric” to the real
heart of geometry. We needed a certain amount of foundation material before we could begin to get into
more unfamiliar, challenging concepts and relationships. We have the definitions and postulates, and analogs
of the equality properties, as the foundation. From here on out, we will be able to experience geometry on
a richer and deeper level.
2.7 Lesson Exercises
Prove the Segment Congruence Properties, in questions 1-3.
1. Reflexive:
2. Symmetric: If
, then
3.Transitive: If
and
, then
4. Is the following statement true? If it’s not, give a counterexample. If it is, prove it.
If
and
, then
+
5. Give a reason for each statement in the proof below.
106
=
+
If
,
Given:
and
are collinear, and
, and
, then
.
are collinear, and
Prove:
6. Is the following statement true? Explain your answer. (A formal two-column proof is not required.)
Let
interior of
and
be points in a single plane. If
, then
is in the interior of
is in the interior of
, and
is in the
.
Note that this is a bit like a Transitive Property for a ray in the interior of an angle.
Answers
1.
Statement
Reason
A.
Reflexive Property of Equality
B.
Definition of congruent angles
2.
Given:
Prove:
Statement
Reason
A.
Given
B.
Definition of congruent angles
C.
Symmetric Property of Equality
D.
Definition of congruent angles
3.
Given:
;
Prove:
Statement
A.
B.
Reason
and
and
Given
Definition of congruent angles
C.
Transitive Property of Equality
D.
Definition of congruent angles
107
4. Yes
Given:
and
Prove:
Statement
Reason
A.
and
B.
,
Given
Definition of congruent angles
C.
Addition Property of Equality
D.
Substitution
5.
Statement
Reason
, and
A._____ Given
are collinear
B._____ Given
C._____ Definition of congruent segments
D._____ Addition Property of Equality
E._____ Commutative Property of Equality
F._____ Definition of collinear points
G._____ Definition collinear points
H._____ Substitution Property of Equality
I._____ Definition of congruent segments
6. True. Since
the interior of
is in the interior of
,
, then
(
. So
)
(
is in the interior of
by the angle addition property.
Proofs About Angle Pairs
Learning Objectives
•
108
Since
State theorems about special pairs of angles.
is in
•
Understand proofs of the theorems about special pairs of angles.
•
Apply the theorems in problem solving.
Introduction
So far most of the things we have proven have been fairly straightforward. Now we have the tools to prove
some more in-depth theorems that may not be so obvious. We’ll start with theorems about special pairs of
angles. They are:
•
right angles
•
supplementary angles
•
complementary angles
•
vertical angles
Right Angle Theorem
If two angles are right angles, then the angles are congruent.
Given:
and
are right angles.
Prove:
Statement
Reason
and
1.
2.
are right angles.
Given
Definition of right angle
,
3.
Substitution
4.
Definition of congruent angles
Supplements of the Same Angle Theorem
If two angles are both supplementary to the same angle (or congruent angles) then the angles are congruent.
Comments: As an example, we know that if
is supplementary to a
angle, then
If
is also supplementary to a
angle, then
too, and
Given:
and
are supplementary angles.
and
.
are supplementary angles.
Prove:
Statement
Reason
1.
and
are supplementary angles. Given
2.
and
are supplementary angles.
3 .
Given
, Definition of supplementary angles
4.
Substitution
5.
Addition Property of Equality
109
Definition of congruent angles
6.
Example 1
Given that
, what other angles must be congruent?
Answer:
by the Right Angle Theorem, because they’re both right angles.
by the Supplements of the Same Angle Theorem and the Linear Pair
and
are a linear pair, which makes them supplementary.
and
Postulate:
are also a linear pair, which makes them supplementary too. Then by Supplements
of the Same Angle Theorem,
because they’re supplementary to congruent
and
.
angles
Complements of the Same Angle Theorem
If two angles are both complementary to the same angle (or congruent angles) then the angles are congruent.
Comments: Only one word is different in this theorem compared to the Supplements of the Same Angle
Theorem. Here we have angles that are complementary, rather than supplementary, to the same angle.
The proof of the Complements of the Same Angle Theorem is in the Lesson Exercises, and it is very similar
to the proof above.
Vertical Angles Theorem
Vertical Angles Theorem: Vertical angles are congruent.
Vertical angles are common in geometry problems, and in real life wherever lines intersect: cables, fence
lines, highways, roof beams, etc. A theorem about them will be useful. The vertical angle theorem is one of
the world’s briefest theorems. Its proof draws on the new theorems just proved earlier in this section.
Given: Lines
Prove:
110
and
, and
intersect.
Statement
1. Lines
Reason
and
Given
intersect.
2.
and
,
3.
and
are supplementary, and
and
Definition of linear pairs
are linear pairs.
and
are supplementary.
Linear Pair Postulate
Supplements of the Same
Angle Theorem
4.
This shows that
. The same proof can be used to show that
.
Example 2
Given:
,
Each of the following pairs of angles are congruent. Give a reason.
and
answer: Vertical Angles Theorem
and
answer: Complements of Congruent Angles Theorem
and
answer: Vertical Angles Theorem
and
answer: Vertical Angles Theorem
and
answer: Vertical Angles Theorem and Transitive Property
and
answer: Vertical Angles Theorem and Transitive Property
and
answer: Complements of Congruent Angles Theorem
Example 3
•
Given:
•
Prove:
,
111
Statement
1.
Reason
,
Given
2.
Vertical Angles Theorem
3.
Transitive Property of Congruence
Lesson Summary
In this lesson we proved theorems about angle pairs.
•
Right angles are congruent.
•
Supplements of the same, or congruent, angles are congruent.
•
Complements of the same, or congruent, angles are congruent.
•
Vertical angles are congruent.
We saw how these theorems can be applied in simple or complex figures.
Points to Consider
Advice to the geometry student:
KISS, or Keep It Simple, Student!
No matter how complicated or abstract the model of a real-world situation may seem, in the final analysis it
can often be expressed in terms of simple lines, segments, and angles. We’ll be able to use the theorems
of this section when we encounter complicated relationships in future figures.
Lesson Exercises
Use the diagram to answer questions 1-3.
Given:
112
Fill in the blanks.
1.
_________
2.
_________
3.
_________
4. Fill in the reasons in the following proof.
Given:
and
Prove:
Statement
Reason
A.___
and
and
are right angles
and
B.___
C.___
D.____
E.___
113
and
mentary
are complementary ,
and
are comple- F.___
G._
5. Which of the following statements must be true? Answer Yes or No.
A.
B.
C.
6. The following diagram shows a ray of light that is reflected from a mirror. The dashed segment is perpen.
dicular to the mirror.
is called the angle of incidence;
is called the angle of reflection. Explain how you know that the
angle of incidence is congruent to the angle of reflection.
Answers
1.
2.
3.
4.
A. Given
B. Definition of perpendicular segments
C. Angle Addition Postulate
114
D. Definition of Right Angle
E. Substitution (Transitive property of equality)
F. Definition of Complementary Angles
G. Complements of the Same Angle are Congruent
5.
A. No
B. Yes
C. No
6.
and
are complementary;
plements of congruent angles and
and
are complementary .
because they are com-
.
115
3. Parallel and Perpendicular Lines
Lines and Angles
Learning Objectives
•
Identify parallel lines, skew lines, and parallel planes.
•
Know the statement of and use the Parallel Line Postulate.
•
Know the statement of and use the Perpendicular Line Postulate.
•
Identify angles made by transversals.
Introduction
In this chapter, you will explore the different types of relationships formed with parallel and perpendicular
lines and planes. There are many different ways to understand the angles formed, and a number of tricks
to find missing values and measurements. Though the concepts of parallel and perpendicular lines might
seem complicated, they are present in our everyday life. Roads are often parallel or perpendicular, as are
crucial elements in construction, such as the walls of a room. Remember that every theorem and postulate
in this chapter can be useful in practical applications.
Parallel and Perpendicular Lines and Planes, and Skew Lines
Parallel lines are two or more lines that lie in the same plane and never intersect.
We use the symbol
for parallel, so to describe the figure above we would write
draw a pair of parallel lines, we use an arrow mark
to show that the lines are parallel. Just like with
congruent segments, if there are two (or more) pairs of parallel lines, we use one arrow
and two (or more) arrows
. When we
for one pair
for the other pair.
Perpendicular lines intersect at a right angle. They form a
by a small square box in the
angle.
angle. This intersection is usually shown
117
The symbol
we could write
is used to show that two lines, segments, or rays are perpendicular. In the preceding picture,
. (Note that
is a ray while
is a line.)
Note that although "parallel" and "perpendicular" are defined in terms of lines, the same definitions apply to
rays and segments with the minor adjustment that two segments or rays are parallel (perpendicular) if the
lines that contain the segments or rays are parallel (perpendicular).
Example 1
Which roads are parallel and which are perpendicular on the map below?
The first step is to remember the definitions or parallel and perpendicular lines. Parallel lines lie in the same
plane but will never intersect. Perpendicular lines intersect at a right angle. All of the roads on this map lie
in the same plane, and Rose Avenue and George Street never intersect. So, they are parallel roads. Henry
Street intersects both Rose Avenue and George Street at a right angle, so it is perpendicular to those roads.
Planes can be parallel and perpendicular just like lines. Remember that a plane is a two-dimensional surface
that extends infinitely in all directions. If planes are parallel, they will never intersect. If they are perpendicular,
they will intersect at a right angle.
118
Two parallel planes
The orange plane and green plane are
both perpendicular to the blue plane.
If you think about a table, the top of the table and the floor below it are usually in parallel planes.
The other of relationship you need to understand is skew lines. Skew lines are lines that are in different
planes, and never intersect. Segments and rays can also be skew. In the cube shown below segment
and segment
are skew. Can you name other pairs of skew segments in this diagram? (How many
pairs of skew segments are there in all?)
Example 2
What is the relationship between the front and side of the building in the picture below?
(Source: http://commons.wikimedia.org/wiki/File:California_Hotel_(Oakland,_CA).JPG, License: Creative
Commons Attribution ShareAlike 2.5)
119
The planes that are represented by the front and side of the building above intersect at the corner. The
corner appears to be a right angle
so the planes are perpendicular.
Parallel Line Postulate
As you already know, there are many different postulates and theorems relating to geometry. It is important
for you to maintain a list of these ideas as they are presented throughout these chapters. One of the postulates
that involves lines and planes is called the Parallel Line Postulate.
Parallel Postulate: Given a line and a point not on the line, there is exactly one line parallel to the given
line that goes through that point. Look at the following diagram to see this illustrated.
Line
in the diagram above is near point
. If you want to draw a line that is parallel to
that goes
through point
there is only one option. Think of lines that are parallel to
as different latitude, like on
, but only one will travel through point
.
a map. They can be drawn anywhere above and below line
Example 3
Draw a line through point
that is parallel to line
.
Remember that there are many different lines that could be parallel to line
There can only be one line parallel to
120
that travels through point
.
. This line is drawn below.
Perpendicular Line Postulate
Another postulate that is relevant to these scenarios is the Perpendicular Line Postulate.
Perpendicular Line Postulate: Given a line and a point not on the line, there is exactly one line perpendicular to the given line that passes through the given point.
This postulate is very similar to the Parallel Line Postulate, but deals with perpendicular lines. Remember
that perpendicular lines intersect at a right
angle. So, as in the diagram below, there is only one line
that can pass through point
while being perpendicular to line .
Example 4
Draw a line through point
that is perpendicular to line
.
Remember that there can only be one line perpendicular to
drawn below.
that travels through point
. This line is
121
Angles and Transversals
Many math problems involve the intersection of three or more lines. Examine the diagram below.
In the diagram, lines
situation:
122
and
are crossed by line
. We have quite a bit of vocabulary to describe this
•
Line is called a transversal because it intersects two other lines (
with and
forms eight angles as shown.
•
and
The area between lines
and
is called the exterior.
•
Angles
and
are called adjacent angles because they share a side and do not overlap. There
and
,
and
, and
and
are many pairs of adjacent angles in this diagram, including
.
•
and
are vertical angles. They are nonadjacent angles made by the intersection of two lines.
Other pairs of vertical angles in this diagram are
and
,
and
, and
and
.
•
Corresponding angles are in the same position relative to both lines crossed by the transversal.
is on the upper left corner of the intersection of lines and .
is on the upper left corner of the intersection of lines
and . So we say that
and
are corresponding angles.
•
and
are called alternate interior angles. They are in the interior region of the lines
and are on opposite sides of the transversal.
•
Similarly,
and
are alternate exterior angles because they are on opposite sides of the
transversal, and in the exterior of the region between and
.
•
Finally,
and
are consecutive interior angles. They are on the interior of the region between
lines and
and are next to each other.
and
are also consecutive interior angles.
and
). The intersection of line
is called the interior of the two lines. The area not between lines
and
Example 5
List all pairs of alternate angles in the diagram below.
There are two types of alternate angles—alternate interior angles and alternate exterior angles. As you
need to list them both, begin with the alternate interior angles.
Alternate interior angles are on the interior region of the two lines crossed by the transversal, so that would
include angles
and
Alternate angles are on opposite sides of the transversal,
pairs of alternate interior angles are
&
, and
and
.
. So, the two
Alternate exterior angles are on the exterior region of the two lines crossed by the transversal, so that would
include angles
and
Alternate angles are on opposite sides of the transversal,
&
, and
and
.
pairs of alternate exterior angles are
. So, the two
Lesson Summary
In this lesson, we explored how to work with different types of lines, angles and planes. Specifically, we
have learned:
•
How to identify parallel lines, skew lines, and parallel planes.
•
How to identify and use the Parallel Line Postulate.
•
How to identify and use the Perpendicular Line Postulate.
•
How to identify angles and transversals of many types.
These will help you solve many different types of problems. Always be on the lookout for new and interesting
ways to examine the relationship between lines, planes, and angles.
Points to Consider
Parallel planes are two planes that do not intersect. Parallel lines must be in the same plane and they do
not intersect. If more than two lines intersect at the same point and they are perpendicular, then they cannot
be in same plane (e.g., the
,
, and
axes are all perpendicular). However, if just two lines are
perpendicular, then there is a plane that contains those two lines.
As you move on in your studies of parallel and perpendicular lines you will usually be working in one plane.
This is often assumed in geometry problems. However, you must be careful about instances where you are
working with multiple planes in space. Generally in three-dimensional space parallel and perpendicular lines
are more challenging to work with.
Lesson Exercises
Solve each problem.
123
1. Imagine a line going through each branch of the tree below (see the red lines in the image). What term
best describes the two branches with lines in the tree pictured below?
(Source: Derived from http://commons.wikimedia.org/wiki/File:Mammutbaum.jpg., License: Public Domain)
2. How many lines can be drawn through point
that will be parallel to line
3. Which of the following best describes skew lines?
A. They lie in the same plane but do not intersect.
B. They intersect, but not at a right angle.
C. They lie in different planes and never intersect.
D. They intersect at a right angle.
4. Are the sides of the Transamerica Pyramid building in San Francisco parallel?
124
(Source: http://commons.wikimedia.org/wiki/File:SF_Transamerica_top_CA.jpg, License: Creative Commons
Attribution ShareAlike 2.5)
5. How many lines can be drawn through point
that will be perpendicular to line
6. Which of the following best describes parallel lines?
A. They lie in the same plane but do not intersect.
B. They intersect, but not at a right angle.
C. They lie in different planes and never intersect.
D. They intersect at a right angle.
7. Draw five parallel lines in the plane. How many regions is the plane divided into by these five lines?
8. If you draw
parallel lines in the plane, how many regions will the plane be divided into?
The diagram below shows two lines cut by a transversal. Use this diagram to answer questions 9 and 10.
125
9. What term best describes the relationship between angles 1 and 5?
A. Consecutive interior
B. Alternate exterior
C. Alternate interior
D. Corresponding
10. What term best describes angles 7 and 8?
A. Linear pair
B. Alternate exterior
C. Alternate interior
D. Corresponding
Answers
1. Skew [Diff: 1]
2. One [Diff: 1]
3. C [Diff: 2]
4. No [Diff: 1]
5. One [Diff: 1]
6. A [Diff: 2]
7. Five parallel lines divide the plane into six regions
126
8.
parallel lines divide the plane into
regions [Diff: 3]
9. D [Diff: 3]
10. A [Diff: 3]
Parallel Lines and Transversals
Learning Objectives
•
Identify angles formed by two parallel lines and a non-perpendicular transversal.
•
Identify and use the Corresponding Angles Postulate.
•
Identify and use the Alternate Interior Angles Theorem.
•
Identify and use the Alternate Exterior Angles Theorem.
•
Identify and use the Consecutive Interior Angles Theorem.
Introduction
In the last lesson, you learned to identify different categories of angles formed by intersecting lines. This
lesson builds on that knowledge by identifying the mathematical relationships inherent within these categories.
Parallel Lines with a Transversal—Review of Terms
As a quick review, it is helpful to practice identifying different categories of angles.
Example 1
In the diagram below, two vertical parallel lines are cut by a transversal.
Identify the pairs of corresponding angles, alternate interior angles, alternate exterior angles, and consecutive
interior angles.
•
Corresponding angles: Corresponding angles are formed on different lines, but in the same relative position to the transversal—in other words, they face the same direction. There are four pairs of corresponding angles in this diagram—
and
,
and
,
and
, and
and
.
127
•
Alternate interior angles: These angles are on the interior of the lines crossed by the transversal and are
on opposite sides of the transversal. There are two pairs of alternate interior angles in this diagram—
and
, and
and
.
•
Alternate exterior angles: These are on the exterior of the lines crossed by the transversal and are on
opposite sides of the transversal. There are two pairs of alternate exterior angles in this diagram—
and
, and
and
.
•
Consecutive interior angles: Consecutive interior angles are in the interior region of the lines crossed by
the transversal, and are on the same side of the transversal. There are two pairs of consecutive interior
angles in this diagram—
and
and
and
.
Corresponding Angles Postulate
By now you have had lots of practice and should be able to easily identify relationships between angles.
Corresponding Angles Postulate: If the lines crossed by a transversal are parallel, then corresponding
angles will be congruent. Examine the following diagram.
You already know that
and
are corresponding angles because they are formed by two lines crossed
by a transversal and have the same relative placement next to the transversal. The Corresponding Angles
postulate says that because the lines are parallel to each other, the corresponding angles will be congruent.
Example 2
In the diagram below, lines
and
are parallel. What is the measure of
?
Because lines
and
are parallel, the
angle and
are corresponding angles, we know by the
Corresponding Angles Postulate that they are congruent. Therefore,
.
Alternate Interior Angles Theorem
Now that you know the Corresponding Angles Postulate, you can use it to derive the relationships between
all other angles formed when two lines are crossed by a transversal. Examine the angles formed below.
128
If you know that the measure of
is
you can find the measurement of all the other angles. For
example,
and
must be supplementary (sum to
) because together they are a linear pair
(we are using the Linear Pair Postulate here). So, to find
, subtract
from
So,
. Knowing that
and
are also supplementary means that
since
. If
, then
must be
because
and
are also supplemen(they both measure
) and
(both measure
). These angles
tary. Notice that
are called vertical angles. Vertical angles are on opposite sides of intersecting lines, and will always be
congruent by the Vertical Angles Theorem, which we proved in an earlier chapter. Using this information,
you can now deduce the relationship between alternate interior angles.
Example 3
Lines
and
in the diagram below are parallel. What are the measures of angles
and
?
In this problem, you need to find the angle measures of two alternate interior angles given an exterior angle.
Use what you know. There is one angle that measures
by the Corresponding Angles Postulate,
Angle
corresponds to the
angle. So
.
Now, because
is made by the same intersecting lines and is opposite the
angle, these two angles
are vertical angles. Since you already learned that vertical angles are congruent, we conclude
. Finally, compare angles
and
. They both measure
time two parallel lines are cut by a transversal.
so they are congruent. This will be true any
We have shown that alternate interior angles are congruent in this example. Now we need to show that it
is always true for any angles.
129
Alternate Interior Angles Theorem Alternate interior angles formed by two parallel lines
and a transversal will always be congruent.
•
•
Given:
and
are parallel lines crossed by transversal
Prove that Alternate Interior Angles are congruent
Note: It is sufficient to prove that one pair of alternate interior angles are congruent. Let's focus on proving
.
Statement
Reason
1.Given
1.
2.
2. Corresponding Angles Postulate
3.
3. Vertical Angles Theorem
4.
4. Transitive property of congruence
Alternate Exterior Angles Theorem
Now you know that pairs of corresponding, vertical, and alternate interior angles are congruent. We will use
logic to show that Alternate Exterior Angles are congruent—when two parallel lines are crossed by a
transversal, of course.
Example 4
Lines
and
in the diagram below are parallel. If
You know from the problem that
, will measure
as well.
130
. That means that
, what is the measure of
?
’s corresponding angle, which is
The corresponding angle you just filled in is also vertical to
can conclude
.
. Since vertical angles are congruent, you
This example is very similar to the proof of the alternate exterior angles Theorem. Here we write out the
theorem in whole:
Alternate Exterior Angles Theorem If two parallel lines are crossed by a transversal,
then alternate exterior angles are congruent.
We omit the proof here, but note that you can prove alternate exterior angles are congruent by following the
method of example 4, but not using any particular measures for the angles.
Consecutive Interior Angles Theorem
The last category of angles to explore in this lesson is consecutive interior angles. They fall on the interior
of the parallel lines and are on the same side of the transversal. Use your knowledge of corresponding angles
to identify their mathematical relationship.
Example 5
Lines
what is
and
in the diagram below are parallel. If the angle corresponding to
measures
?
This process should now seem familiar. The given
Therefore, the angles are supplementary. So, to find
angle is adjacent to
and they form a linear pair.
, subtract
from
This example shows that if two parallel lines are cut by a transversal, the consecutive interior angles are
supplementary; they sum to
This is called the Consecutive Interior Angles Theorem. We restate it
here for clarity.
131
Consecutive Interior Angles Theorem If two parallel lines are crossed by a transversal,
then consecutive interior angles are supplementary.
Proof: You will prove this as part of your exercises.
Lesson Summary
In this lesson, we explored how to work with different angles created by two parallel lines and a transversal.
Specifically, we have learned:
•
How to identify angles formed by two parallel lines and a non-perpendicular transversal.
•
How to identify and use the Corresponding Angles Postulate.
•
How to identify and use the Alternate Interior Angles Theorem.
•
How to identify and use the Alternate Exterior Angles Theorem.
•
How to identify and use the Consecutive Interior Angles Theorem.
These will help you solve many different types of problems. Always be on the lookout for new and interesting
ways to analyze lines and angles in mathematical situations.
Points To Consider
You used logic to work through a number of different scenarios in this lesson. Always apply logic to mathematical situations to make sure that they are reasonable. Even if it doesn’t help you solve the problem, it
will help you notice careless errors or other mistakes.
Lesson Exercises
Solve each problem.
Use the diagram below for Questions 1-4. In the diagram, lines
1. What term best describes the relationship between
and
are parallel.
and
a. alternate exterior angles
b. consecutive interior angles
c. corresponding angles
d. alternate interior angles
2. What term best describes the mathematical relationship between
132
and
?
a. congruent
b. supplementary
c. complementary
d. no relationship
3. What term best describes the relationship between
and
a. alternate exterior angles
b. consecutive interior angles
c. complementary
d. alternate interior angles
4. What term best describes the mathematical relationship between
and
a. congruent
b. supplementary
c. complementary
d. no relationship
Use the diagram below for questions 5-7. In the diagram, lines
measures of the angles.
and
are parallel
represent the
5. What is
6. What is
7. What is
The map below shows some of the streets in Ahmed’s town.
133
Jimenez Ave and Ella Street are parallel. Use this map to answer questions 8-10.
8. What is the measure of angle 1?
9. What is the measure of angle 2?
10. What is the measure of angle 3?
11. Prove the Consecutive Interior Angle Theorem. Given
Answers
1. C [Diff: 1]
2. A [Diff: 2]
3. D [Diff: 2]
4. B [Diff: 2]
5.
134
[Diff: 1]
6.
[Diff: 2]
7.
[Diff: 2]
, prove
and
are supplementary.
8.
[Diff: 2]
9.
[Diff: 2]
10.
[Diff: 2]
11. Proof of Consecutive Interior Angle Theorem. Given
, prove
Statement
Reason
1.
s
2.
2. Corresponding Angles Postulate
3.
and
are supplementary 3. Linear Pair Postulate
4. Definition of supplementary angles
5.
5. Substitution (
and
are supplementary.
1
.
Given
4.
6.
and
)
are supplementary 6. Definition of supplementary angles
Proving Lines Parallel
Learning Objectives
•
Identify and use the Converse of the Corresponding Angles Postulate.
•
Identify and use the Converse of Alternate Interior Angles Theorem.
•
Identify and use the Converse of Alternate Exterior Angles Theorem.
•
Identify and use the Converse of Consecutive Interior Angles Theorem.
•
Identify and use the Parallel Lines Property.
Introduction
If two angles are vertical angles, then they are congruent. You learned this as the Vertical Angles Theorem.
Can you reverse this statement? Can you swap the “if” and “then” parts and will the statement still be true?
The converse of a logical statement is made by reversing the hypothesis and the conclusion in an if-then
statement. With the Vertical Angles Theorem, the converse is “If two angles are congruent then they are
vertical angles.” Is that a true statement? In this case, no. The converse of the Vertical Angles Theorem is
135
NOT true. There are many examples of congruent angles that are not vertical angles—for example the
corners of a square.
Sometimes the converse of an if-then statement will also be true. Can you think of an example of a statement
in which the converse is true? This lesson explores converses to the postulates and theorems about parallel
lines and transversals.
Corresponding Angles Converse
Let’s apply the concept of a converse to the Corresponding Angles Postulate. Previously you learned that
"if two parallel lines are cut by a transversal, the corresponding angles will be congruent." The converse of
this statement is "if corresponding angles are congruent when two lines are cut by a transversal, then the
two lines crossed by the transversal are parallel." This converse is true, and it is a postulate.
Converse of Corresponding Angles Postulate If corresponding angles are congruent
when two lines are crossed by a transversal, then the two lines crossed by the transversal
are parallel.
Example 1
Suppose we know that
?
Notice that
and
and
. What can we conclude about lines
are corresponding angles. Since
Corresponding Angles Postulate and conclude that
and
, we can apply the Converse of the
.
You can also use converse statements in combination with more complex logical reasoning to prove whether
lines are parallel in real life contexts. The following example shows a use of the contrapositive of the Corresponding Angles Postulate.
Example 2
The three lines in the figure below represent metal bars and a cable supporting a water tower.
136
(Source: Derived from http://commons.wikimedia.org/wiki/File:Amarillo-Texas-Water-Tower-Dec2005.jpg,
License: Creative Commons Attribution ShareAlike 2.5)
and
. Are the lines
and
parallel?
To find out whether lines
and
are parallel, you must identify the corresponding angles and see if they
are congruent. In this diagram,
and
are corresponding angles because they are formed by the
transversal and the two lines crossed by the transversal and they are in the same relative place.
The problem states that
and
. Thus, they are not congruent. If those two
angles are not congruent, the lines are not parallel. In this scenario, the lines
and
(and thus the
support bars they represent) are NOT parallel.
Note that just because two lines may look parallel in the picture that is not enough information to say that
the lines are parallel. To prove two lines are parallel you need to look at the angles formed by a transversal.
Alternate Interior Angles Converse
Another important theorem you derived in the last lesson was that when parallel lines are cut by a
transversal, the alternate interior angles formed will be congruent. The converse of this theorem is, "If alternate
interior angles formed by two lines crossed by a transversal are congruent, then the lines are parallel." This
statement is also true, and it can be proven using the Converse of the Corresponding Angles Postulate.
Converse of Alternate Interior Angles Theorem If two lines are crossed by a
transversal and alternate interior angles are congruent, then the lines are parallel.
137
Given
and
are crossed by
and
.
Prove
Statement
1.
and
Reason
are crossed by
.
and
1. Given
2.
2. Vertical Angles Theorem
3.
3. Transitive Property of Angle Congruence
4.
4. Converse of the Corresponding Angles
Postulate.
Notice in the proof that we had to show that the corresponding angles were congruent. Once we had done
that, we satisfied the conditions of the Converse of the Corresponding Angles postulate, and we could use
that in the final step to prove that the lines are parallel.
Example 3
Are the two lines in this figure parallel?
This figure shows two lines that are cut by a transversal. We don't know
. However, if you look at its
linear pair, that angle has a measure of
By the Linear Pair Postulate, this angle is supplementary to
. In other words, the sum of
and
will be
Use subtraction to find
.
So,
. Now look and
Angles Theorem,
.
138
.
is a vertical angle with the angle measuring
By the Vertical
Since \angle{1} \cong \angle{2} as can apply the converse of the Alternate Interior Angles Theorem to
conclude that
.
Notice in this example that you could have also used the Converse of the Corresponding Angles Postulate
to prove the two lines are parallel. Also, This example highlights how, if a figure is not drawn to scale you
cannot assume properties of the objects in the figure based on looks.
Converse of Alternate Exterior Angles
The more you practice using the converse of theorems to find solutions, the easier it will become. You have
already probably guessed that the converse of the Alternate Exterior Angles Theorem is true.
Converse of the Alternate Exterior Angles Theorem If two lines are crossed by a
transversal and the alternate exterior angles are congruent, then the lines crossed by the
transversal are parallel.
Putting together the alternate exterior angles theorem and its converse, we get the biconditional statement:
Two lines crossed by a transversal are parallel if and only if alternate exterior angles are congruent.
Use the example below to apply this concept to a real-world situation.
Example 4
The map below shows three roads in Julio’s town.
In Julio's town, Franklin Way and Chavez Avenue are both crossed by Via La Playa. Julio used a surveying
tool to measure two angles in the intersections as shown and he drew the sketch above (NOT to scale).
Julio wants to know if Franklin Way is parallel to Chavez Avenue. How can he solve this problem and what
is the correct answer?
Notice that this question asks you not only to identify the answer, but also the process required to solve it.
Make sure that your solution is step-by-step so that anyone reading it can follow your logic.
To begin, notice that the labeled
angle and
are alternate exterior angles. If these two angles are
congruent, then the lines are parallel. If they are not congruent, the lines are not parallel. To find the measure
of angle
, you can use the other angle labeled in this diagram, measuring
This angle is supplementary to
because they are a linear pair. Using the knowledge that a linear pair must be supplementary,
find the value of
.
139
Angle
is equal to
This angle is
wider than the other alternate exterior angle, which
measures
so the alternate exterior angles are not congruent. Therefore, Franklin Way and Chavez
Avenue are not parallel streets.
In this example, we used the contrapositive of the converse of the Alternate Exterior Angles Theorem to
prove that the two lines were not parallel.
Converse of Consecutive Interior Angles
The final converse theorem to explore in this lesson addressed the Consecutive Interior Angles Theorem.
Remember that these angles aren’t congruent when lines are parallel, they are supplementary. In other
words, if the two lines are parallel, the angles on the interior and on the same side of the transversal will
sum to
So, if two consecutive interior angles made by two lines and a transversal add up to
the two lines that form the consecutive angles are parallel.
Example 5
Identify whether lines
and
in the diagram below are parallel.
Using the converse of the Consecutive Interior Angles Theorem, you should be able to identify that if the
are parallel. We add the two consecutive
two angles in the figure are supplementary, then lines and
interior angles to find their sum.
?
The two angles in the figure sum to
so lines
and
are in fact parallel.
Parallel Lines Property
The last theorem to explore in this lesson is called the Parallel Lines Property. It is a transitive property.
Does the phrase transitive property sound familiar? You have probably studied other transitive properties
before, but usually talking about numbers. Examine the statement below.
If
and
, then
Notice that we used a property similar to the transitive property in a proof above. The Parallel Lines Property
says that if line is parallel to line
, and line
is parallel to line
, then lines and
are also parallel. Use this information to solve the final practice problem in this lesson.
Example 6
Are lines
140
and
in the diagram below parallel?
Look at this diagram carefully to establish the relationship between lines
and
and lines
and
.
Starting with line , the angle shown measures
This angle is an alternate exterior angle to the
angle labeled on line . Since the alternate exterior angles are congruent, these two lines are parallel. Next
look at the relationship between and . The angle shown on line measures
and it corresponds
to the
angle marked on line . Since the corresponding angles on these two lines are congruent, lines
and are also parallel.
Using the Parallel Lines Property, we can identify that lines
and is also parallel to .
and
are parallel, because
is parallel to
Note that there are many other ways to reason through this problem. Can you think of one or two alternative
ways to show
?
Lesson Summary
In this lesson, we explored how to work with the converse of theorems we already knew. Specifically, we
have learned:
•
How to identify and use the Corresponding Angles Converse Postulate.
•
How to identify and use the Converse of Alternate Interior Angles Theorem.
•
How to identify and use the Converse of Alternate Exterior Angles Theorem.
•
How to identify and use the Converse of Consecutive Interior Angles Theorem.
•
How to identify and use the Parallel Lines Property.
These will help you solve many different types of problems. Always be on the lookout for new and interesting
ways to apply theorems and postulates to mathematical situations.
Points To Consider
You have now studied the many rules about parallel lines and the angles they form. In the next lesson, you
will delve deeper into concepts of lines in the
-plane. You will apply some of the geometric properties
of lines to slopes and graphing in the coordinate plane.
Lesson Exercises
Solve each problem.
1. Are lines
and
parallel in the diagram below? If yes, how do you know?
141
2. Are lines 1 and 2 parallel in the following diagram? Why or why not?
3. Are lines
142
and
parallel in the diagram below? Why or why not?
4. Are lines
and
parallel in the following diagram? Justify your answer.
5. Are lines
and
parallel in the diagram below? Justify your answer.
143
For exercises 6-13, use the following diagram. Line
cation for each of your answers.
6.
144
________.
and
. Find each angle and give a justifi-
7.
________.
8.
________.
9.
________.
10.
________.
11.
________.
12.
________.
13.
________.
Answers
1. Yes. If alternate interior angles are congruent, then the lines are parallel [Diff: 1].
2. No. Since alternate exterior angles are NOT congruent the lines are NOT parallel [Diff: 1].
3. No. Since alternate interior angles are NOT congruent, the lines are NOT parallel. [Diff: 1].
4. Yes. If corresponding angles are congruent, then the lines are parallel [Diff: 1].
5. Yes. If exterior angles on the same side of the transversal are supplementary, then the lines are parallel
[Diff: 2].
6.
. Since
7.
and the
.
8.
9.
10.
and
are a linear pair, they are supplementary [Diff: 2].
is an interior angle on the same side of transversal
angle are supplementary and
[Diff: 3].
.
is a vertical angle to the angle marked
. It is a corresponding angle with angle
So
[Diff 2]
[Diff: 3].
. It is a linear pair with a right angle [Diff: 1].
11.
. It is a corresponding angle with the angle marked
12.
. It is a vertical angle with
13.
with the angle marked
[Diff: 3].
[Diff: 2].
. It is a linear pair with the angle marked
[Diff: 2].
145
4. Congruent Triangles
Triangle Sums
Learning Objectives
•
Identify interior and exterior angles in a triangle.
•
Understand and apply the Triangle Sum Theorem.
•
Utilize the complementary relationship of acute angles in a right triangle.
•
Identify the relationship of the exterior angles in a triangle.
Introduction
In the first chapter of this course, you developed an understanding of basic geometric principles. The rest
of this course explores specific ideas, techniques, and rules that will help you be a successful problem solver.
If you ever want to review the basic problem solving in geometry return to Chapter 1. This chapter explores
triangles in more depth. In this lesson, you’ll explore some of their basic components.
Interior and Exterior Angles
Any closed structure has an inside and an outside. In geometry we use the words interior and exterior for
the inside and outside of a figure. An interior designer is someone who furnishes or arranges objects inside
a house or office. An external skeleton (or exo-skeleton) is on the outside of the body. So the prefix “ex”
means outside and exterior refers to the outside of a figure.
The terms interior and exterior help when you need to identify the different angles in triangles. The three
angles inside the triangles are called interior angles. On the outside, exterior angles are the angles formed
by extending the sides of the triangle. The exterior angle is the angle formed by one side of the triangle and
the extension of the other.
197
Note: In triangles and other polygons there are TWO sets of exterior angles, one “going” clockwise, and the
other “going” counterclockwise. The following diagram should help.
But, if you look at one vertex of the triangle, you will see that the interior angle and an exterior angle form
a linear pair. Based on the Linear Pair Postulate, we can conclude that interior and exterior angles at the
same vertex will always be supplementary. This tells us that the two exterior angles at the same vertex are
congruent.
Example 1
What is
in the triangle below?
The question asks for
. The exterior angle at vertex
and exterior angles sum to
measures
Since interior
you can set up an equation.
interior angle + exterior angle =
Thus,
.
Triangle Sum Theorem
Probably the single most valuable piece of information regarding triangles is the Triangle Sum Theorem.
198
Triangle Sum Theorem The sum of the measures of the interior angles in a triangle is
180°
Regardless of whether the triangle is right, obtuse, acute, scalene, isosceles, or equilateral, the interior angles
Examine each of the triangles shown below.
will always add up to
Notice that each of the triangles has an angle that sums to
You can also use the triangle sum theorem to find a missing angle in a triangle. Set the sum of the angles
and solve for the missing value.
equal to
Example 2
What is
in the triangle below?
Set up an equation where the three angle measures sum to
Then, solve for
.
Now that you have seen an example of the triangle sum theorem at work, you may wonder, why it is true.
The answer is actually surprising: The measures of the angles in a triangle add to
because of the
199
Parallel line Postulate. Here is a proof of the triangle sum theorem.
•
Given:
•
Prove:
as in the diagram below,
that
the
measures
of
the
three
angles
add
to
or
in
symbols,
that
.
Statement
1. Given
2. Through point
to
Reason
1. Given
in the diagram
, draw the line parallel 2. Parallel Postulate
. We will call it
3. Alternate interior Angles Theorem
3.
4. Alternate interior Angles Theorem
4.
5.
5. Angle Addition postulate
6.
6. Linear Pair Postulate
7.
7. Substitution (also known as “transitive
property of equality”)
8.
8. Substitution (Combining steps 3, 4, and
7).
And that proves that the sum of the measures of the angles in ANY triangle is
Acute Angles in a Right Triangle
Expanding on the triangle sum theorem, you can find more specific relationships. Think about the implications
of the triangle sum theorem on right triangles. In any right triangle, by definition, one of the angles is a right
angle—it will always measure
resulting in a total sum of
This means that the sum of the other two angles will always be
Therefore the two acute angles in a right triangle will always be complementary and as one of the angles
gets larger, the other will get smaller so that their sum is 90°.
200
Recall that a right angle is shown in diagrams by using a small square marking in the angle, as shown below.
So, when you know that a triangle is right, and you have the measure of one acute angle, you can easily
find the other.
Example 3
What is the measure of the missing angle in the triangle below?
Since the triangle above is a right triangle, the two acute angles must be complementary. Their sum will be
We will represent the missing angle with the variable and write an equation.
Now we can use inverse operations to isolate the variable, and then we will have the measure of the
missing angle.
The measure of the missing angle is
Exterior Angles in a Triangle
One of the most important lessons you have learned thus far was the triangle sum theorem, stating that the
sum of the measure of the interior angles in any triangle will be equal to
You know, however, that
there are two types of angles formed by triangles: interior and exterior. It may be that there is a similar theorem that identifies the sum of the exterior angles in a triangle.
Recall that the exterior and interior angles around a single vertex sum to
as shown below.
201
Imagine an equilateral triangle and the exterior angles it forms. Since each interior angle measures
each exterior angle will measure
What is the sum of these three angles? Add them to find out.
The sum of these three angles is
In fact, the sum of the exterior angles in any triangle will always be
You can use this information just as you did the triangle sum theorem to find missing angles
equal to
and measurements.
Example 4
What is the value of
in the triangle below?
You can set up an equation relating the three exterior angles to
Remember that does not represent
an exterior angle, so do not use that variable. Solve for the value of the exterior angle. Let's call the measure
of the exterior angle .
202
The missing exterior angle measures
You can use this information to find the value of
the interior and exterior angles form a linear pair and therefore they must sum to
, because
Exterior Angles in a Triangle Theorem In a triangle, the measure of an exterior angle
is equal to the sum of the remote interior angles.
We won’t prove this theorem with a two-column proof (that will be an exercise), but we will use the example
above to illustrate it. Look at the diagram from the previous example for a moment. If we look at the exterior
angle at
, then the interior angles at
and
are called “remote interior angles.”
Notice that the exterior angle at point
measured
measured
and the interior angle at
At the same time, the interior angle at point
measured
The sum of interior angles
. Notice the measures of the remote interior angles sum to the
measure of the exterior angle at
. This relationship is always true, and it is a result of the linear pair
postulate and the triangle sum theorem. Your job will be to show how this works.
Lesson Summary
In this lesson, we explored triangle sums. Specifically, we have learned:
•
How to identify interior and exterior angles in a triangle.
•
How to understand and apply the Triangle Sum Theorem
•
How to utilize the complementary relationship of acute angles in a right triangle.
•
How to identify the relationship of the exterior angles in a triangle.
203
These skills will help you understand triangles and their unique qualities. Always look for triangles in diagrams,
maps, and other mathematical representations.
Points to Consider
Now that you understand the internal qualities of triangles, it is time to explore the basic concepts of triangle
congruence.
Lesson Exercises
Questions 1 and 2 use the following diagram:
1. Find
2. What is
in the triangle above.
in the triangle above?
Questions 3-6 use the following diagram:
3. What is
4. What is
5. What is
6. What is the relationship between
know this is the relationship.
7. Find
204
in the diagram below:
and
? Write one or two sentences to explain how you
Use the diagram below for questions 8-13. (Note
8.
_____. Why?
9.
_____. Why?
10.
_____. Why?
11.
_____. Why?
12.
_____. Why?
13.
_____. Why?
)
14. Prove the Remote Exterior Angle Theorem: The measure of an exterior angle in a triangle equals the
sum of the measures of the remote interior angles. To get started, you may use the following: Given triangle
as in the diagram below, prove
.
Answers
1.
[Diff: 1]
2.
[Diff: 1]
3.
[Diff: 1]
4.
[Diff: 1]
5.
[Diff: 2]
6.
and
add up to
are complementary. Since the measures of the three angles of the triangle must
we can use the fact that
is a right angle to conclude that
[Diff: 2].
205
7.
[Diff: 2]
8.
and
9.
.
10.
[Diff:3].
is an alternate interior angle with the labeled
.
12.
[Diff: 2].
is an alternate interior angle with
.
11.
add up to
is a linear pair with
[Diff:3].
[Diff:3].
. Use the triangle sum theorem with
13.
. Use the triangle sum theorem with
and solve for
[Diff:3].
[Diff:3].
14. We will prove this using a two-column proof.
Statement
Reason
1. Given
1.
2.
2. Triangle Sum Theorem
3.
3. Linear Pair Postulate
4.
4. Substitution
5.
5. Subtraction property of equality (subtracted
sides)
on both
Congruent Figures
Learning Objectives
206
•
Define congruence in triangles.
•
Create accurate congruence statements.
•
Understand that if two angles of a triangle are congruent to two angles of another triangle, the remaining
angles will also be congruent.
•
Explore properties of triangle congruence.
Introduction
Triangles are important in geometry because every other polygon can be turned into triangles by cutting
them up (formally we call this adding auxiliary lines). Think of a square: If you add an auxiliary line such
as a diagonal, then it is two right triangles. If we understand triangles well, then we can take what we know
about triangles and apply that knowledge to all other polygons. In this chapter you will learn about congruent
triangles, and in subsequent chapters you will use what you know about triangles to prove things about all
kinds of shapes and figures.
Defining Congruence in Triangles
Two figures are congruent if they have exactly the same size and shape. Another way of saying this is that
the two figures can be perfectly aligned when one is placed on top of the other—but you may need to rotate
or flip the figures over to make them line up. When that alignment is done, the angles that are matched are
called corresponding angles, and the sides that are matched are called corresponding sides.
In the diagram above, sides
and
have the same length, as shown by the tic marks. If two sides
have the same number of tic marks, it means that they have the same length. Since
have one tic mark, they have the same length. Once we have established that
examine the other sides of the triangles.
and
and
each
, we need to
each have two tic marks, showing that they are also
because they each have three tic marks. Each of these
congruent. Finally, as you can see,
pairs corresponds because they are congruent to each other. Notice that the three sides of each triangle
do not need to be congruent to each other, as long as they are congruent to their corresponding side on the
other triangle.
When two triangles are congruent, the three pairs of corresponding angles are also congruent. Notice the
tic marks in the triangles below.
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We use arcs inside the angle to show congruence in angles just as tic marks show congruence in sides.
From the markings in the angles we can see
and
.
By definition, if two triangles are congruent, then you know that all pairs of corresponding sides are congruent
and all pairs of corresponding angles are congruent. This is sometimes called CPCTC: Corresponding parts
of congruent triangles are congruent.
Example 1
Are the two triangles below congruent?
The question asks whether the two triangles in the diagram are congruent. To identify whether or not the
triangles are congruent, each pair of corresponding sides and angles must be congruent.
Begin by examining the sides.
and
both have one tic mark, so they are congruent.
both have two tic marks, so they are congruent as well.
pair of sides is congruent.
Next you must check each angle.
and
because they each have two arcs. Finally,
and
and
have three tic marks each, so each
both have one arc, so they are congruent.
because they have three arcs.
We can check that each angle in the first triangle matches with its corresponding angle in the second triangle
by examining the sides.
corresponds with
because they are formed by the sides with two and
three tic marks. Since all pairs of corresponding sides and angles are congruent in these two triangles, we
conclude that the two triangles are congruent.
Creating Congruence Statements
We have already been using the congruence sign
angles.
For example, if you wanted to say that
when talking about congruent sides and congruent
was congruent to
, you could write the following statement.
In Chapter 1 you learned that the line above
with no arrows means that
is a segment (and not
a line or a ray). If you were to read this statement out loud, you could say “Segment
is congruent to
segment
.”
When dealing with congruence statements involving angles or triangles, you can use other symbols. Whereas
means “segment
the symbol
means “triangle
.”
,” the symbol
means “angle
.” Similarly, the symbol
When you are creating a congruence statement of two triangles, the order of the letters is very important.
Corresponding parts must be written in order. That is, the angle at first letter of the first triangle corresponds with the angle at the first letter of the second triangle, the angles at the second letter correspond,
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and so on.
In the diagram above, if you were to name each triangle individually, they could be
and
. Those names seem the most appropriate because the letters are in alphabetical order. However, if you
are writing a congruence statement, you could NOT say that
. If you look at
, it
does not correspond to
.
corresponds to
instead (indicated by the two arcs in the angles).
corresponds to
, and
corresponds to
. Remember, you must compose the congruent
statement so that the vertices are lined up for congruence. The statement below is correct.
This form may look strange at first, but this is how you must create congruence statements in any situation.
Using this standard form allows your work to be easily understood by others, a crucial element of mathematics.
Example 2
Compose a congruence statement for the two triangles below.
To write an accurate congruence statement, you must be able to identify the corresponding pairs in the triangles above. Notice that
and
each have one arc mark. Similarly,
and
each have two
arcs, and
and
have three arcs. Additionally,
(or
),
and
So, the two triangles are congruent, and to make the most accurate statement, this should be expressed
by matching corresponding vertices. You can spell the first triangle in alphabetical order and then align the
second triangle to that standard.
Notice in example 2 that you don’t need to write the angles in alphabetical order, as long as corresponding
parts match up. If you’re feeling adventurous, you could also express this statement as shown below.
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Both of these congruence statements are accurate because corresponding sides and angles are aligned
within the statement.
The Third Angle Theorem
Previously, you studied the triangle sum theorem, which states that the sum of the measures of the interior
angles in a triangle will always be equal to
This information is useful when showing congruence. As
you practiced, if you know the measures of two angles within a triangle, there is only one possible measurement of the third angle. Thus, if you can prove two corresponding angle pairs congruent, the third pair is
also guaranteed to be congruent.
Third Angle Theorem If two angles in one triangle are congruent to two angles in another
triangle, then the third pair of angles are also congruent.
This may seem like an odd statement, but use the exercise below to understand it more fully.
Example 3
Identify whether or not the missing angles in the triangles below are congruent.
To identify whether or not the third angles are congruent, you must first find their measures. Start with the
triangle on the left. Since you know two of the angles in the triangle, you can use the triangle sum theorem
to find the missing angle. In
we know
The missing angle of the triangle on the left measures
Repeat this process for the triangle on the right.
So,
. Remember that you could also identify this without using the triangle sum theorem. If
two pairs of angles in two triangles are congruent, then the remaining pair of angles also must be congruent.
Congruence Properties
In earlier mathematics courses, you have learned concepts like the reflexive or commutative properties.
These concepts help you solve many types of mathematics problems. There are a few properties relating
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to congruence that will help you solve geometry problems as well.
The reflexive property of congruence states that any shape is congruent to itself. This may seem obvious,
but in a geometric proof, you need to identify every possibility to help you solve a problem. If two triangles
share a line segment, you can prove congruence by the reflexive property.
In the diagram above, you can say that the shared side of the triangles is congruent because of the reflexive
property. Or in other words,
.
The symmetric property of congruence states that congruence works frontwards and backwards, or in
symbols, if
then
.
The transitive property of congruence states that if two shapes are congruent to a third, they are also
congruent to each other. In other words, if
, and
, then
. This property is very important in identifying congruence between different shapes.
Example 4
Which property can be used to prove the statement below?
and
If
, then
.
A. reflexive property of congruence
B. identity property of congruence
C. transitive property of congruence
D. symmetric property of congruence
The transitive property is the one that allows you to transfer congruence to different shapes. As this states
that two triangles are congruent to a third, they must be congruent to each other by the transitive property.
The correct answer is C.
Lesson Summary
In this lesson, we explored congruent figures. Specifically, we have learned:
•
How to define congruence in triangles.
•
How to create accurate congruence statements.
•
To understand that if two angles of a triangle are congruent to two angles of another triangle, the remaining
angles will also be congruent.
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•
How to employ properties of triangle congruence.
These skills will help you understand issues of congruence involving triangles. Always look for triangles in
diagrams, maps, and other mathematical representations.
Points to Consider
Now that you understand the issues inherent in triangle congruence, you will create your first congruence
proof.
Lesson Exercises
Use the diagram below for problem 1.
1. Write a congruence statement for the two triangles above.
Exercises 2-3 use the following diagram.
2. Suppose the two triangles above are congruent. Write a congruence statement for these two triangles.
3. Explain how we know that if the two triangles are congruent, then
.
Use the diagram below for exercises 4-5.
4. Explain how we know
.
5. Are these two triangles congruent? Explain why (note, “looks” are not enough of a reason!).
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6. If you want to know the measure of all three angles in a triangle, how many angles do you need to measure
with your protractor? Why?
Use the following diagram for exercises 7-10.
7. What is the relationship between
and
? How do you know?
? How do you know?
8. What is
9. What property tells us
?
10. Write a congruence statement for these triangles.
Answers
1.
[Diff: 1]
2.
(Note the order of the letters is important!) [Diff:2].
3. If the two triangles are congruent, then
each other by the definition of congruence.
corresponds with
and therefore they are congruent to
4. The third angle theorem states that if two pairs of angles are congruent in two triangles, then the third
pair of angles must also be congruent [Diff: 1].
5. No.
corresponds with
but they are not the same length [Diff: 2].
6. You only need to measure two angles. The triangle sum theorem will help you find the measure of the
third angle [Diff: 2].
7.
8.
and
are supplementary since they are a linear pair [Diff: 2].
[Diff: 3].
9. The reflexive property of congruence [Diff: 3].
10.
[Diff: 3].
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Triangle Congruence using SSS
Learning Objectives
•
Use the distance formula to analyze triangles on a coordinate grid.
•
Understand and apply the SSS postulate of triangle congruence.
Introduction
In the last section you learned that if two triangles are congruent then the three pairs of corresponding sides
are congruent and the three pairs of corresponding angles are congruent. In symbols,
means
, and
.
Wow, that’s a lot of information—in fact, one triangle congruence statement contains six different congruence
statements! In this section we show that proving two triangles are congruent does not necessarily require
showing all six congruence statements are true. Lucky for us, there are shortcuts for showing two triangles
are congruent—this section and the next explore some of these shortcuts.
Triangles on a Coordinate Grid
To begin looking at rules of triangle congruence, we can use a coordinate grid. The following grid shows
two triangles.
The first step in finding out if these triangles are congruent is to identify the lengths of the sides. In algebra,
you learned the distance formula, shown below.
You can use this formula to find the distances on the grid.
Example 1
Find the distances of all the line segments on the coordinate grid above using the distance formula.
Begin with
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. First write the coordinates.
is
is
is
Now use the coordinates to find the lengths of each segment in the triangle.
So, the lengths are as follows.
,
, and
Next, find the lengths in triangle
. First write the coordinates.
is
is
is
Now use the coordinates to find the lengths of each segment in the triangle.
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So, the lengths are as follows:
,
, and
Using the distance formula, we showed that the corresponding sides of the two triangles have the same
length. We don’t have the tools to find the measures of the angles in these triangles, so we can show congruence in a different way.
without changing its shape and move the whole triangle units right
Imagine you could pick up
and units down. If you did this, then points
and
would be on top of each other,
and
would
also be on top of each other, and
and
would also coincide.
To analyze the relationship between the points, the distance formula is not necessary. Simply look at how
far (and in what direction) the vertices may have moved.
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Points
and
:
is
Points
and
:
is
Points
and
:
is
and
is
and
and
.
is
is
.
.
is
is
is
units to the right and
units to the right and
units to the right and
units below
.
units below
units below
.
.
Since the same relationship exists between the vertices, you could move the entire triangle
units
to the right and units down. It would exactly cover triangle
. These triangles are therefore congruent.
SSS Postulate of Triangle Congruence
The extended example above illustrates that when three sides of one triangle are equal in length to three
sides of another, then the triangles are congruent. We did not need to measure the angles—the lengths of
the corresponding sides being the same “forced” the corresponding angles to be congruent. This leads us
to one of the triangle congruence postulates:
Side-Side-Side (SSS) Triangle Congruence Postulate: If three sides in one triangle
are congruent to the three corresponding sides in another triangle, then the triangles are
congruent to each other.
This is a postulate so we accept it as true without proof.
You can perform a quick experiment to test this postulate. Cut two pieces of spaghetti (or a straw, or some
segment-like thing) exactly the same length. Then cut another set of pieces that are the same length as
each other (but not necessarily the same length as the first set). Finally, cut one more pair of pieces of
spaghetti that are identical to each other. Separate the pieces into two piles. Each pile should have three
pieces of different lengths. Build a triangle with one set and leave it on your desk. Using the other pieces,
attempt to make a triangle with a different shape or size by matching the ends. Notice that no matter what
you do, you will always end up with congruent triangles (though they might be “flipped over” or rotated). This
demonstrates that if you can identify three pairs of congruent sides in two triangles, the two triangles are
fully congruent.
Example 2
Write a triangle congruence statement based on the diagram below:
We can see from the tick marks that there are three pairs of corresponding congruent sides:
,
, and
. Matching up the corresponding sides, we can write the congruence statement
.
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Don’t forget that ORDER MATTERS when writing triangle congruence statements. Here, we lined up the
sides with one tic mark, then the sides with two tic marks, and finally the sides with three tic marks.
Lesson Summary
In this lesson, we explored triangle congruence using only the sides. Specifically, we have learned:
•
How to use the distance formula to analyze triangles on a coordinate grid
•
How to understand and apply the SSS postulate of triangle congruence.
These skills will help you understand issues of congruence involving triangles, and later you will apply this
knowledge to all types of shapes.
Points to Consider
Now that you have been exposed to the SSS Postulate, there are other triangle congruence postulates to
explore. The next chapter deals with congruence using a mixture of sides and angles.
Lesson Exercises
1. If you know that
in the diagram below, what are six congruence statements that
you also know about the parts of these triangles?
2. Redraw these triangles using geometric markings to show all congruent parts.
Use the diagram below for exercises 3-7 .
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3. Find the length of each side in
a.
b.
c.
4. Find the length of each side in
a.
b.
c.
5. Write a congruence statement relating these two triangles.
6. Write another equivalent congruence statement for these two triangles.
7. What postulate guarantees these triangles are congruent?
Exercises 8-10 use the following diagram:
8. Write a congruence statement for the two triangles in this diagram. What postulate did you use?
9. Find
10. Find
. Explain how you know your answer.
. Explain how you know your answer.
Answers
1.
and
[Diff: 1]
2. One possible answer: [Diff: 1]
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3. a.
units, b.
4. a.
units, b.
units, c.
units [Diff: 2]
units, c.
units [Diff: 2]
5.
[Diff: 2]
(Note, other answers are possible, but the relative order of the letters does matter.)
6.
[Diff: 3]
7. SSS [Diff: 2]
, the side-side-side triangle congruence postulate [Diff: 3]
8.
. We know this because it corresponds with
9.
10.
, so
[Diff: 3]
. Used the triangle sum theorem together with my answer for 9. [Diff: 3]
Triangle Congruence Using ASA and AAS
Learning Objectives
•
Understand and apply the ASA Congruence Postulate.
•
Understand and apply the AAS Congruence Theorem.
•
Understand and practice two-column proofs.
•
Understand and practice flow proofs.
Introduction
The SSS Congruence Postulate is one of the ways in which you can prove two triangles are congruent
without measuring six angles and six sides. The next two lessons explore other ways in which you can prove
triangles congruent using a combination of sides and angles. It is helpful to know all of the different ways
you can prove congruence between two triangles, or rule it out if necessary.
ASA Congruence
One of the other ways you can prove congruence between two triangles is the ASA Congruence Postulate.
The “S” represents “side,” as it did in the SSS Theorem. “A” stands for “angle” and the order of the letters
in the name of the postulate is crucial in this circumstance. To use the ASA postulate to show that two triangles
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are congruent, you must identify two angles and the side in between them. If the corresponding sides and
angles are congruent, the entire triangles are congruent. In formal language, the ASA postulate is this:
Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side
in one triangle are congruent to two angles and the included side in another triangle, then
the two triangles are congruent.
To test out this postulate, you can use a ruler and a protractor to make two congruent triangles. Start by
drawing a segment that will be one side of your first triangle and pick two angles whose sum is less than
Draw one angle on one side of the segment, and draw the second angle on the other side. Now, repeat
the process on another piece of paper, using the same side length and angle measures. What you’ll find is
that there is only one possible triangle you could create—the two triangles will be congruent.
Notice also that by picking two of the angles of the triangle, you have determined the measure of the third
by the Triangle Sum Theorem. So, in reality, you have defined the whole triangle; you have identified all of
the angles in the triangle, and by picking the length of one side, you defined the scale. So, no matter what,
if you have two angles, and the side in between them, you have described the whole triangle.
Example 1
What information would you need to prove that these two triangles are congruent using the ASA postulate?
A. the measures of the missing angles
B. the measures of sides
and
C. the measures of sides
and
D. the measures of sides
and
If you are to use the ASA postulate to prove congruence, you need to have two pairs of congruent angles
and the included side, the side in between the pairs of congruent angles. The side in between the two marked
angles in
is side
. The side in between the two marked angles in
is side
You
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would need the measures of sides
and
to prove congruence. The correct answer is C.
AAS Congruence
Another way you can prove congruence between two triangles is using two angles and the non-included
side.
Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side
in one triangle are congruent to two corresponding angles and a non-included side in
another triangle, then the triangles are congruent.
This is a theorem because it can be proven. First, we will do an example to see why this theorem is true,
then we will prove it formally. Like the ASA postulate, the AAS theorem uses two angles and a side to prove
triangle congruence. However, the order of the letters (and the angles and sides they stand for) is different.
The AAS theorem is equivalent to the ASA postulate because when you know the measure of two angles
in a triangle, you also know the measure of the third angle. The pair of congruent sides in the triangles will
determine the size of the two triangles.
Example 2
What information would you need to prove that these two triangles were congruent using the AAS theorem?
A. the measures of sides
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and
B. the measures of sides
C. the measures of
D. the measures of angles
and
and
and
If you are to use the AAS theorem to prove congruence, you need to know that pairs of two angles are
congruent and the pair of sides adjacent to one of the given angles are congruent. You already have one
side and its adjacent angle, but you still need another angle. It needs to be the angle not touching the known
side, rather than adjacent to it. Therefore, you need to find the measures of
and
to prove
congruence. The correct answer is D.
When you use AAS (or any triangle congruence postulate) to show that two triangles are congruent, you
need to make sure that the corresponding pairs of angles and sides actually align. For instance, look at the
diagram below:
Even though two pairs of angles and one pair of sides are congruent in the triangles, these triangles are
NOT congruent. Why? Notice that the marked side in
is
which is between the unmarked
angle and the angle with two arcs. However in
, the marked side is between the unmarked angle
and the angle with one arc. As the corresponding parts do not match up, you cannot use AAS to say these
triangles are congruent.
AAS and ASA
The AAS triangle congruence theorem is logically the exact same as the ASA triangle congruence postulate.
Look at the following diagrams to see why.
Since
and
, we can conclude from the third angle theorem that
.
This is because the sum of the measures of the three angles in each triangle is
and if we know the
measures of two of the angles, then the measure of the third angle is already determined. Thus, marking
, the diagram becomes this:
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Now we can see that
(A),
(S), and
(A), which shows that
by ASA.
Proving Triangles Congruent
In geometry we use proofs to show something is true. You have seen a few proofs already—they are a
special form of argument in which you have to justify every step of the argument with a reason. Valid reasons
are definitions, postulates, or results from other proofs.
One way to organize your thoughts when writing a proof is to use a two-column proof. This is probably the
most common kind of proof in geometry, and it has a specific format. In the left column you write statements
that lead to what you want to prove. In the right hand column, you write a reason for each step you take.
Most proofs begin with the “given” information, and the conclusion is the statement you are trying to prove.
Here’s an example:
Example 3
Create a two-column proof for the statement below.
Given:
is the bisector of
, and
Prove:
Remember that each step in a proof must be clearly explained. You should formulate a strategy before you
begin the proof. Since you are trying to prove the two triangles congruent, you should look for congruence
between the sides and angles. You know that if you can prove SSS, ASA, or AAS, you can prove congruence.
Since the given information provides two pairs of congruent angles, you will most likely be able to show the
triangles are congruent using the ASA postulate or the AAS theorem. Notice that both triangles share one
side. We know that side is congruent to itself
, and now you have pairs of two congruent
angles and non-included sides. You can use the AAS congruence theorem to prove the triangles are con224
gruent.
Statement
1.
Reason
is the bisector of
1. Given
2. Definition of an angle bisector (a bisector divides
an angle into two congruent angles)
2.
3. Given
3.
4. Reflexive Property
4.
5. AAS Congruence Theorem (if two pairs of angles
and the corresponding non-included sides are congruent, then the triangles are congruent)
5.
Notice how the markings in the triangles help in the proof. Whenever you do proofs, use arcs in the angles
and tic marks to show congruent angles and sides.
Flow Proofs
Though two-column proofs are the most traditional style (in geometry textbooks, at least!), there are many
different ways of solving problems in geometry. We already wrote a paragraph proof in an earlier lesson
that simply described, step by step, the rationale behind an assertion (when we showed why AAS is logically
equivalent to ASA). The two-column style is easy to read and organizes ideas clearly. Some students,
however, prefer flow proofs. Flow proofs show the relationships between ideas more explicitly by using a
chart that shows how one idea will lead to the next. Like two-column proofs, it is helpful to always remember
the end goal so you can identify what it is you need to prove. Sometimes it is easier to work backwards!
The next example repeats the same proof as the one above, but displayed in a flow style, rather than two
columns.
Example 4
Create a flow proof for the statement below.
Given:
is the bisector of
and
Prove:
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As you can see from these two proofs of the theorem, there are many different ways of expressing the same
information. It is important that you become familiar with proving things using all of these styles because
you may find that different types of proofs are better suited for different theorems.
Lesson Summary
In this lesson, we explored triangle congruence. Specifically, we have learned to:
•
Understand and apply the ASA Congruence Postulate.
•
Understand and apply the AAS Congruence Postulate.
•
Understand and practice Two-Column Proofs.
•
Understand and practice Flow Proofs.
These skills will help you understand issues of congruence involving triangles. Always look for triangles in
diagrams, maps, and other mathematical representations.
Points to Consider
Now that you have been exposed to the SAS and AAS postulates, there are even more triangle congruence
postulates to explore. The next lesson deals with SAS and HL proofs.
Lesson Exercises
Use the following diagram for exercises 1-3.
1. Complete the following congruence statement, if possible
________.
2. What postulate allows you to make the congruence statement in 1, or, if it is not possible to make a congruence statement explain why.
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3. Given the marked congruent parts, what other congruence statements do you now know based on your
answers to 1 and 2?
Use the following diagram for exercises 4-6 .
4. Complete the following congruence statement, if possible
_______.
5. What postulate allows you to make the congruence statement in 4, or, if it is not possible to make a congruence statement explain why.
6. Given the marked congruent parts in the triangles above, what other congruence statements do you now
know based on your answers to 4 and 5?
Use the following diagram for exercises 7-9.
7. Complete the following congruence statement, if possible
________.
8. What postulate allows you to make the congruence statement in 7, or, if it is not possible to make a congruence statement explain why.
9. Given the marked congruent parts in the triangles above, what other congruence statements do you now
know based on your answers to 7 and 8?
10. Complete the steps of this two-column proof:
Given
, and
Prove:
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Note: You cannot assume that
and
are collinear or that
Statement
Reason
1.
1. Given
2.
2. _______
3. _______
3. Given
4.
and
are collinear.
4. _______ triangle congruence postulate
_____
5. _______________________________
5.
11. Bonus question: Why do we have to use three letters to name
use only one letter to name
or
?
and
, while we can
Answers
1.
[Diff: 1]
2. AAS triangle congruence postulate [Diff: 1]
3.
and
[Diff: 2]
4. No congruence statement is possible [Diff: 3]
5. We can’t use either AAS or ASA because the corresponding parts do not match up [Diff: 3]
6.
3]
. This is still true by the third angle theorem, even if the triangles are not congruent. [Diff:
7.
[Diff: 1]
8. ASA triangle congruence postulate [Diff: 1]
9.
, and
[Diff: 2]
10. [Diff: 3]
Statement
Reason
1.
1. Given
2.
2. Given
3.
3. Given
4.
5.
4. AAS Triangle Congruence Postulate
5. Definition of congruent triangles (if two
then all corresponding parts
triangles are
are also
).
11. We can use one letter to name an angle when there is no ambiguity. So at point
in the diagram for
10 there is only one possible angle. At point
there are four angles, so we use the “full name” of the angles
to be specific! [Diff: 2]
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Proof Using SAS and HL
Learning Objectives
•
Understand and apply the SAS Congruence Postulate.
•
Identify the distinct characteristics and properties of right triangles.
•
Understand and apply the HL Congruence Theorem.
•
Understand that SSA does not necessarily prove triangles are congruent.
Introduction
You have already seen three different ways to prove that two triangles are congruent (without measuring
six angles and six sides). Since triangle congruence plays such an important role in geometry, it is important
to know all of the different theorems and postulates that can prove congruence, and it is important to know
which combinations of sides and angles do not prove congruence.
SAS Congruence
By now, you are very familiar with postulates and theorems using the letters
and
to represent triangle
sides and angles. One more way to show two triangles are congruent is by the SAS Congruence Postulate.
SAS Triangle Congruence Postulate: If two sides and the included angle in one triangle
are congruent to two sides and the included angle in another triangle, then the two triangles
are congruent.
Like ASA and AAS congruence, the order of the letters is very significant. You must have the angles between
the two sides for the SAS postulate to be valid.
Once again you can test this postulate using physical models (such as pieces of uncooked spaghetti) for
the sides of a triangle. You’ll find that if you make two pairs of congruent sides, and lay them out with the
same included angle then the third side will be determined.
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Example 1
What information would you need to prove that these two triangles were congruent using the SAS postulate?
A. the measures of
and
B. the measures of
and
C. the measures of
D. the measures of sides
and
and
If you are to use the SAS postulate to establish congruence, you need to have the measures of two sides
and the angle in between them for both triangles. So far, you have one side and one angle. So, you must
use the other side adjacent to the same angle. In
corresponding side is
, that side is
. In triangle
, the
. So, the correct answer is C.
AAA and SSA relationships
You have learned so many different ways to prove congruence between two triangles, it may be tempting
to think that if you have any pairs of congruent three elements (combining sides or angles), you can prove
triangle congruence.
However, you may have already guessed that AAA congruence does not work. Even if all of the angles are
equal between two triangles, the triangles may be of different scales. So, AAA can only prove similarity,
not congruence.
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SSA relationships do not necessarily prove congruence either. Get your spaghetti and protractors back on
your desk to try the following experiment. Choose two pieces of spaghetti at given length. Select a measure
for an angle that is not between the two sides. If you keep that angle constant, you may be able to make
two different triangles. As the angle in between the two given sides grows, so does the side opposite it. In
other words, if you have two sides and an angle that is not between them, you cannot prove congruence.
In the figure,
is NOT congruent to
and a pair of congruent angles.
that can be made using this combination SSA.
even though they have two pairs of congruent sides
and you can see that there are two possible triangles
Example 2
Can you prove that the two triangles below are congruent?
Note: Figure is not to scale.
The two triangles above look congruent, but are labeled, so you cannot assume that how they look means
that they are congruent. There are two sides labeled congruent, as well as one angle. Since the angle is
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not between the two sides, however, this is a case of SSA. You cannot prove that these two triangles are
congruent. Also, it is important to note that although two of the angles appear to be right angles, they are
not marked that way, so you cannot assume that they are right angles.
Right Triangles
So far, the congruence postulates we have examined work on any triangle you can imagine. As you know,
there are a number of types of triangles. Acute triangles have all angles measuring less than
Obtuse
triangles have one angle measuring between
and
Equilateral triangles have congruent sides,
and all angles measure
Right triangles have one angle measuring exactly
In right triangles, the sides have special names. The two sides adjacent to the right angle are called legs
and the side opposite the right angle is called the hypotenuse.
Example 3
Which side of right triangle
is the hypotenuse?
Looking at
, you can identify
as a right angle (remember the little square tells us the
angle is a right angle). By definition, the hypotenuse of a right triangle is opposite the right angle. So, side
is the hypotenuse.
HL Congruence
There is one special case when SSA does prove that two triangles are congruent-When the triangles you
are comparing are right triangles. In any two right triangles you know that they have at least one pair of
congruent angles, the right angles.
Though you will learn more about it later, there is a special property of right triangles referred to as the
Pythagorean theorem. It isn’t important for you to be able to fully understand and apply this theorem in
this context, but it is helpful to know what it is. The Pythagorean Theorem states that for any right triangle
with legs that measure and and hypotenuse measuring units, the following equation is true.
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In other words, if you know the lengths of two sides of a right triangle, then the length of the third side can
be determined using the equation. This is similar in theory to how the Triangle Sum Theorem relates angles.
You know that if you have two angles, you can find the third.
Because of the Pythagorean Theorem, if you know the length of the hypotenuse and a leg of a right triangle,
you can calculate the length of the missing leg. Therefore, if the hypotenuse and leg of one right triangle
are congruent to the corresponding parts of another right triangle, you could prove the triangles congruent
by the SSS congruence postulate. So, the last in our list of theorems and postulates proving congruence is
called the HL Congruence Theorem. The “H” and “L” stand for hypotenuse and leg.
HL Congruence Theorem: If the hypotenuse and leg in one right triangle are congruent
to the hypotenuse and leg in another right triangle, then the two triangles are congruent.
The proof of this theorem is omitted because we have not yet proven the Pythagorean Theorem.
Example 4
What information would you need to prove that these two triangles were congruent using the HL theorem?
A. the measures of sides
and
B. the measures of sides
and
C. the measures of angles
and
D. the measures of angles
and
Since these are right triangles, you only need one leg and the hypotenuse to prove congruence. Legs
and
are congruent, so you need to find the lengths of the hypotenuses. The hypotenuse of
is
. The hypotenuse of
. The correct answer is A.
is
. So, you need to find the measures of sides
and
Points to Consider
The HL congruence theorem shows that sometimes SSA is sufficient to prove that two triangles are congruent.
You have also seen that sometimes it is not. In trigonometry you will study this in more depth. For now, you
might try playing with objects or you may try using geometric software to explore under which conditions
SSA does provide enough information to infer that two triangles are congruent.
Lesson Summary
In this lesson, we explored triangle sums. Specifically, we have learned:
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•
How to understand and apply the SAS Congruence Postulate.
•
How to identify the distinct characteristics and properties of right triangles.
•
How to understand and apply the HL Congruence Theorem.
•
That SSA does not necessarily prove triangles are congruent.
These skills will help you understand issues of congruence involving triangles. Always look for triangles in
diagrams, maps, and other mathematical representations.
Lesson Exercises
Use the following diagram for exercises 1-3.
1. Complete the following congruence statement, if possible
_________.
2. What postulate allows you to make the congruence statement in 1, or, if it is not possible to make a congruence statement explain why.
3. Given the marked congruent parts in the triangles above, what other congruence statements do you now
know based on your answers to 1 and 2?
Use the following diagram below for exercises 4-6 .
4. Complete the following congruence statement, if possible
_________.
5. What postulate allows you to make the congruence statement in 4, or, if it is not possible to make a congruence statement explain why.
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6. Given the marked congruent parts in the triangles above, what other congruence statements do you now
know based on your answers to 4 and 5?
Use the following diagram below for exercises 7-9.
7. Complete the following congruence statement, if possible
________.
8. What postulate allows you to make the congruence statement in 7, or, if it is not possible to make a congruence statement explain why.
9. Given the marked congruent parts in the triangles above, what other congruence statements do you now
know based on your answers to 7 and 8?
10. Write one or two sentences and use a diagram to show why AAA is not a triangle congruence postulate.
11. Do the following proof using a two-column format.
Given:
and
intersect at
, and
Prove:
Statement
1.
Reason
1. Given
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2. (Finish the proof using more steps!)
2.
Answers
1.
[Diff: 1]
2. HL triangle congruence postulate [Diff: 2]
3.
and
4.
[Diff: 3]
[Diff: 1]
5. SAS triangle congruence postulate [Diff: 2]
6.
[Diff: 3]
7. No triangle congruence statement is possible [Diff: 2].
8. SSA is not a valid triangle congruence postulate [Diff: 2].
9. No other congruence statements are possible [Diff: 3].
10. One counterexample is to consider two equiangular triangles. If AAA were a valid triangle congruence
postulate, than all equiangular (and equilateral) triangles would be congruent. But this is not the case. Below
are two equiangular triangles that are not congruent: [Diff: 2]
These triangles are not congruent.
11. [Diff: 3]
Statement
Reason
1. Given
1.
2. Given
2.
3.
4.
gles
5.
6.
7.
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and
intersect at
and
are vertical an-
3. Given
4. Definition of vertical angles
5. Vertical angles theorem
6. SAS triangle congruence postulate
7. Definition of congruent triangles (CPCTC)
Using Congruent Triangles
Learning Objectives
•
Apply various triangles congruence postulates and theorems.
•
Know the ways in which you can prove parts of a triangle congruent.
•
Find distances using congruent triangles.
•
Use construction techniques to create congruent triangles.
Introduction
As you can see, there are many different ways to prove that two triangles are congruent. It is important to
know all of the different way that can prove congruence, and it is important to know which combinations of
sides and angles do not prove congruence. When you prove properties of polygons in later chapters you
will frequently use
Congruence Theorem Review
As you have studied in the previous lessons, there are five theorems and postulates that provide different
ways in which you can prove two triangles congruent without checking all of the angles and all of the sides.
It is important to know these five rules well so that you can use them in practical applications.
Name Corresponding congruent parts
Does it prove congruence?
SSS
Three sides
Yes
SAS
Two sides and the angle between them
Yes
ASA
Two angles and the side between them
Yes
AAS
Two angles and a side not between them Yes
HL
A hypotenuse and a leg in a right triangle Yes
AAA
Three angles
SSA
Two sides and an angle not between them No—this can create more than one distinct triangle
No—it will create a similar triangle, but not of the same
size
When in doubt, think about the models we created. If you can construct only one possible triangle given the
constraints, then you can prove congruence. If you can create more than one triangle within the given information, you cannot prove congruence.
Example 1
What rule can prove that the triangles below are congruent?
A. SSS
B. SSA
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C. ASA
D. AAS
The two triangles in the picture have two pairs of congruent angles and one pair of corresponding congruent
sides. So, the triangle congruence postulate you choose must have two
’s (for the angles) and one
(for the side). You can eliminate choices
and
for this reason. Now that you are deciding between
choices
and
, you need to identify where the side is located in relation to the given angles. It is adjacent
to one angle, but it is not in between them. Therefore, you can prove congruence using AAS. The correct
answer is D.
Proving Parts Congruent
It is one thing to identify congruence when all of the important identifying information is provided, but
sometimes you will have to identify congruent parts on your own. You have already practiced this in a few
different ways. When you were testing SSS congruence, you used the distance formula to find the lengths
of sides on a coordinate grid. As a review, the distance formula is shown below.
You can use the distance formula whenever you are examining shapes on a coordinate grid.
When you were creating two-column and flow proofs, you also used the reflexive property of congruence.
This property states that any segment or angle is congruent to itself. While this may sound obvious, it can
be very helpful in proofs, as you saw in those examples.
You may be tempted to use your ruler and protractor to check whether two triangles are congruent. However,
this method does not necessarily work because not all pictures are drawn to scale.
Example 2
in the diagram below?
How could you prove
We can already see that
and
. We may be able to use SSS or SAS to show the
triangles are congruent. However, to use SSS, we would need
and there is no obvious way
to prove this. Can we show that two of the angles are congruent? Notice that
and
are
vertical angles (nonadjacent angles made by the intersection of two lines—i.e., angles on the opposite sides
of the intersection).
The Vertical Angle Theorem states that all vertical angles are also congruent. So, this tells us that
. Finally, by putting all the information together, you can confirm that
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by the SAS Postulate.
Finding Distances
One way to use congruent triangles is to help you find distances in real life—usually using a map or a diagram
as a model.
When using congruent triangles to identify distances, be sure you always match up corresponding sides.
The most common error on this type of problem involves matching two sides that are not corresponding.
Example 3
The map below shows five different towns. The town of Meridian was given its name because it lies exactly
halfway between two pairs of cities: Camden and Grenata, and Lowell and Morsetown.
Using the information in the map, what is the distance between Camden and Lowell?
The first step in this problem is to identify whether or not the marked triangles are congruent. Since you
know that the distance from Camden to Meridian is the same as Meridian to Grenata, those two sides are
congruent. Similarly, since the distance from Lowell to Meridian is the same as Meridian to Morsetown,
those two sides are also a congruent pair. The angles between these lines are also congruent because they
are vertical angles.
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So, by the SAS postulate, these two triangles are congruent. This allows us to find the distance between
Camden and Lowell by identifying its corresponding side on the other triangle. Because they are both opposite
the vertical angle, the side connecting Camden and Lowell corresponds to the side connecting Morsetown
and Grenata. Since the triangles are congruent, these corresponding sides will also be congruent to each
other. Therefore, the distance between Camden and Lowell is five miles.
This use of the definition of congruent triangles is one of the most powerful tools you will use in geometry
class. It is often abbreviated as CPCTC, meaning Corresponding Parts of Congruent Triangles are Congruent.
Constructions
Another important part of geometry is creating geometric figures through construction. A construction is a
drawing that is made using only a straightedge and a compass—you can think of construction as a special
game in geometry in which we make figures using only these tools. You may be surprised how many shapes
can be made this way.
Example 4
Use a compass and straightedge to construct the perpendicular bisector of the segment below.
Begin by using your compass to create an arc with the same distance from a point as the segment.
240
Repeat this process on the opposite side.
Now draw a line through the two points of intersections. This forms the perpendicular bisector.
Draw segments connecting the points on the bisector to the original endpoints.
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Knowing that the center point is the midpoint of both line segments and that all angles formed around point
are right angles, you can prove that all four triangles created are congruent by the SAS rule.
Lesson Summary
In this lesson, we explored applications triangle congruence. Specifically, we have learned to:
•
Identify various triangles congruence postulates and theorems.
•
Use the fact that corresponding parts of congruent triangles are congruent.
•
Find distances using congruent triangles.
•
Use construction techniques to create congruent triangles.
These skills will help you understand issues of congruence involving triangles. Always look for triangles in
diagrams, maps, and other mathematical representations.
Points to Consider
You now know all the different ways in which you can prove two triangles congruent. In the next chapter
you’ll learn more about isosceles and equilateral triangles.
Lesson Exercises
Use the following diagram for exercises 1-5
242
1. Find
in the diagram above.
2. Find
in the diagram above.
3. What is
? How do you know?
4. What postulate can you use to show
5. Use the distance formula to find
. How can use this to find
6-8: For each pair of triangles, complete the triangle congruence statement, or write “no congruence statement
possible.” Name the triangle congruence postulate you use, or write a sentence to explain why you can’t
write a triangle congruence statement.
6.
__________.
7.
__________.
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8.
__________.
9. In the following diagram, Midtown is exactly halfway between Uptown and Downtown. What is the distance
between Downtown and Lower East Side? How do you know? Write a few sentences to convince a reader
your answer is correct.
10. Given:
Prove:
244
is the midpoint of
and
Answers
1.
[Diff: 1]
2.
3.
Since
is horizontal (parallel to the
can conclude that they intersect at a right angle.
-axis) and
is vertical (parallel to the
-axis), we
4. SAS (other answers are possible) [Diff: 2]
. Since the triangles are congruent, we
5.
can conclude
[Diff: 3].
6.
. SAS triangle congruence postulate [Diff: 2]
. SSS triangle congruence postulate [Diff: 2]
7.
8. No congruence statement is possible; we don’t have enough information.
9.
km. Since Midtown is the midpoint of the line connecting Uptown and Downtown, we can use the
vertical angle theorem for the angles made by the two lines that meet at Midtown, and then we can conclude
that the triangles are congruent using AAS. If the triangles are congruent then all corresponding parts are
also congruent [Diff: 3].
10. [Diff: 3]
Statement
Reason
1.
1. Given
2.
3.
is the midpoint of
2. Definition of midpoint
3. Given
4.
4. Alternate interior angles theorem
5.
5. Alternate interior angles theorem
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6. AAS triangle congruence postulate
6.
7. Definition of congruent triangles (corresponding parts are congruent)
7.
Isosceles and Equilateral Triangles
Learning Objectives
•
Prove and use the Base Angles Theorem.
•
Prove that an equilateral triangle must also be equiangular.
•
Use the converse of the Base Angles Theorem.
•
Prove that an equiangular triangle must also be equilateral.
Introduction
As you can imagine, there is more to triangles than proving them congruent. There are many different ways
to analyze the angles and sides within a triangle to understand it better. This chapter addresses some of
the ways you can find information about two special triangles.
Base Angles Theorem
An isosceles triangle is defined as a triangle that has at least two congruent sides. In this lesson you will
prove that an isosceles triangle also has two congruent angles opposite the two congruent sides. The congruent sides of the isosceles triangle are called the legs of the triangle. The other side is called the base
and the angles between the base and the congruent sides are called base angles. The angle made by the
two legs of the isosceles triangle is called the vertex angle.
The Base Angles Theorem states that if two sides of a triangle are congruent, then their opposite angles
are also congruent. In other words, the base angles of an isosceles triangle are congruent. Note, this theorem
does not tell us about the vertex angle.
Example 1
Which two angles must be congruent in the diagram below?
246
The triangle in the diagram is an isosceles triangle. To find the congruent angles, you need to find the angles
that are opposite the congruent sides.
This diagram shows the congruent angles. The congruent angles in the triangle are
.
and
So, how do we prove the base angles theorem? Using congruent triangles.
Given: Isosceles
with
Prove
Statement
1.
Reason
is isosceles with
1. Given
.
2. Construct Angle Bisector
2. Angle Bisector Postulate
3.
3. Definition of Angle Bisector
4.
4. Reflexive Property
5.
5. SAS Postulate
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6.
6. Definition of congruent triangles (all pairs
of corresponding angles are congruent)
Equilateral Triangles
The base angles theorem also applies to equilateral triangles. By definition, all sides in an equilateral triangle have exactly the same length.
Because of the base angles theorem, we know that angles opposite congruent sides in an isosceles triangle
are congruent. So, if all three sides of the triangle are congruent, then all of the angles are congruent as
well.
A triangle that has all angles congruent is called an equiangular triangle. So, as a result of the base angles
theorem, you can identify that all equilateral triangles are also equiangular triangles.
Converse of the Base Angles Theorem
As you know, some theorems have a converse that is also true. Recall that a converse identifies the
“backwards,” or reverse statement of a theorem. For example, if I say, “If I turn a faucet on, then water comes
out,” I have made a statement. The converse of that statement is, “If water comes out of a faucet, then I
have turned the faucet on.” In this case the converse is not true. For example the faucet may have a drip.
So, as you can see, converse statements are sometimes true, but not always.
The converse of the base angles theorem is always true. The base angles theorem states that if two sides
of a triangle are congruent the angles opposite them are also congruent. The converse of this statement is
that if two angles in a triangle are congruent, then the sides opposite them will also be congruent. You can
use this information to identify isosceles triangles in many different circumstances.
Example 2
Which two sides must be congruent in the diagram below?
has two congruent angles. By the converse of the base angles theorem, it is an isosceles triangle.
To find the congruent sides, you need to find the sides that are opposite the congruent angles.
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This diagram shows arrows pointing to the congruent sides. The congruent sides in this triangle are
and
.
The proof of the converse of the base angles theorem will depend on a few more properties of isosceles
triangles that we will prove later, so for now we will omit that proof.
Equiangular Triangles
Earlier in this lesson, you extrapolated that all equilateral triangles were also equiangular triangles and
proved it using the base angles theorem. Now that you understand that the converse of the base angles
theorem is also true, the converse of the equilateral/equiangular relationship will also be true.
If a triangle has three congruent angles, it is be equiangular. Since congruent angles have congruent sides
opposite them, all sides in an equiangular triangle will also be congruent. Therefore, every equiangular
triangle is also equilateral.
Lesson Summary
In this lesson, we explored isosceles, equilateral, and equiangular triangles. Specifically, we have learned
to:
•
Prove and use the Base Angles Theorem.
•
Prove that an equilateral triangle must also be equiangular.
•
Use the converse of the Base Angles Theorem.
•
Prove that an equiangular triangle must also be equilateral.
These skills will help you understand issues of analyzing triangles. Always look for triangles in diagrams,
maps, and other mathematical representations.
Lesson Exercises
1. Sketch and label an isosceles
with legs
2. What is the measure of each base angle in
and
that has a vertex angle measuring
from 1?
3. Find the measure of each angle in the triangle below:
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4.
below is equilateral. If
bisects
, find:
a.
b.
c.
5. Which of the following statements must be true about the base angles of an isosceles triangle ?
a. The base angles are congruent.
b. The base angles are complementary.
c. The base angles are acute.
d. The base angles can be right angles.
6. One of the statements in 5 is possible (i.e., sometimes true), but not necessarily always true. Which one
is it? For the statement that is always false draw a sketch to show why.
7-13: In the diagram below,
each of the following angles.
250
. Use the given angle measure and the geometric markings to find
7.
_____
8.
_____
9.
_____
10.
_____
11.
_____
12.
_____
13.
_____
Answers
1. [Diff: 1]
o
2. Each base angle in
3.
4. a.
measures 31 [Diff: 1]
and
[Diff: 1]
, b.
c.
[Diff: 2]
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5. a. and c. only. [Diff: 3]
6. b. is possible if the base angles are
When this happens, the vertex angle is
d. is impossible
because if the base angles are right angles, then the “sides” will be parallel and you won’t have a triangle.
[Diff: 3]
7.
[Diff: 2]
8.
[Diff: 2]
9.
[Diff: 2]
10.
[Diff: 2]
11.
[Diff: 2]
12.
[Diff: 2]
13.
[Diff: 2]
Congruence Transformations
Learning Objectives
•
Identify and verify congruence transformations.
•
Identify coordinate notation for translations.
•
Identify coordinate notation for reflections over the axes.
•
Identify coordinate notation for rotations about the origin.
Introduction
Transformations are ways to move and manipulate geometric figures. Some transformations result in
congruent shapes, and some don’t. This lesson helps you explore the effect of transformations on congruence
252
and find location of the resulting figures. In this section we will work with figures in the coordinate grid.
Congruence Transformations
Congruent shapes have exactly the same size and shape. Many types of transformations will keep shapes
congruent, but not all. A quick review of transformations follows.
Transformation
Diagram
Congruent or Not?
Translation (Slide)
Congruent
Reflection (Flip)
Congruent
Rotation (Turn)
Congruent
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Dilation
Shrink)
(Enlarge
or
Not Congruent
As you can see, the only transformation in this list that interferes with the congruence of the shapes is dilation.
Dilated figures (whether larger or smaller) have the same shape, but not the same size. So, these shapes
will be similar, but not congruent.
When in doubt, check the length of each side of a triangle in the coordinate grid by using the distance formula.
Remember that if triangles have three pairs of congruent sides, the triangles are congruent by the SSS triangle congruence postulate.
Example 1
Use the distance formula to prove that the reflected image below is congruent to the original triangle
.
Begin with triangle
. First write the coordinates.
is
is
is
Now use the coordinates and the distance formula to find the lengths of each segment in the triangle.
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The lengths are as follows.
,
, and
Next find the lengths in triangle
. First write down the coordinates.
is
is
is
Now use the coordinates to find the lengths of each segment in the triangle.
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The lengths are as follows.
,
, and
Using the distance formula, we demonstrated that the corresponding sides of the two triangles have the
same lengths. Therefore, by the SSS congruence postulate, these triangles are congruent. This example
shows that reflected figures are congruent.
Translations
The transformation you saw above is called a reflection. Translations are another type of transformation.
You translate a figure by moving it right or left and up or down. It is important to know how a transformation
of a figure affects the coordinates of its vertices. You’ll now have the opportunity to practice translating images
and changing the coordinates.
For each unit a figure is translated to the right, add unit to each
-coordinate in the vertices. For each
unit a figure is translated to the left, subtract from the
-coordinates. Always remember that moving a
figure left and right only affects the
-coordinates.
-coordinate. So, if you move a figure up by unit, then
If a figure is translated up or down, it affects the
add unit to each of the
-coordinates in the vertices. Similarly, if you translate a figure down by unit,
subtract unit from the
-coordinates.
Example 2
is shown on the coordinate grid below. What would be the coordinates of
if it has been translated
units to the left and
units up?
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Analyze the change and think about how that will affect the coordinates of the vertices. The translation
moves the figure units to the left. That means you will subtract from each of the
-coordinates. It also
says you will move the figure up units, which means that you will add to each of the
-coordinates.
So, the coordinate change can be expressed as follows.
Carefully adjust each coordinate using the formula above.
This gives us the new coordinates
,
, and
.
Finally, draw the translated triangle to verify that your answer is correct.
Reflections
Reflections are another form of transformation that also result in congruent figures. When you “flip” a figure
over the -axis or -axis, you don’t actually change the shape at all. To find the coordinates of a reflected
figure, use the opposite of one of the coordinates.
257
1. If you reflect an image over the
-axis, the new
-coordinates will be the opposite of the old
-coordinates. The
-coordinates remain the same.
2. If you reflect an image over the
-axis, take the opposite of the
-coordinates. The
-coordinates remain the same.
Example 3
Triangle
is shown on the coordinate grid below. What would be the coordinates of
-axis?
has been reflected over the
if it
Since you are finding the reflection of the image over the
-axis, you will find the opposite of the
-coordinates. The
-coordinates will remain the same. So, the coordinate change can be expressed as follows.
Carefully adjust each coordinate using the formula above.
This gives new coordinates
,
and
Draw the translated triangle to verify that your answer is correct.
258
.
Rotations
The most complicated of the congruence transformations is rotations. To simplify rotations, we will only be
concerned with rotations of
or
about the origin (0,0). The rules describe how coordinates change
under rotations.
rotations: Take the opposite of both coordinates.
becomes
clockwise rotations: Find the opposite of the
-coordinate, and reverse the coordinates.
becomes
counterclockwise rotations: Find the opposite of the.
-coordinate, and reverse the coordinates.
becomes
Example 4
Triangle
is shown on the following coordinate grid. What would be the coordinates of
has been rotated
if it
counterclockwise about the origin?
259
Since you are finding the rotation of the image
counterclockwise about the origin, you will find the opposite of the
-coordinates and then reverse the order. So, the coordinate change can be expressed as
follows.
Carefully adjust each coordinate using the formula above.
This gives new coordinates
,
, and
.
Finally, we draw the rotated triangle to verify that your answer is correct.
260
Lesson Summary
In this lesson, we explored transformations with triangles. Specifically, we have learned to:
•
Identify and verify congruence transformations.
•
Identify coordinate notation for translations.
•
Identify coordinate notation for reflections over the axes.
•
Identify coordinate notation for rotations about the origin.
These skills will help you understand many different situations involving coordinate grids. Always look for
triangles in diagrams, maps, and other mathematical representations.
Lesson Exercises
Use the following diagram of
find the new coordinates of
for exercise 1-4. Given the coordinates
and
after each transformation.
1. Slide down three units.
261
2. Slide up
units and to the right
3. Reflect across the
4. Rotate
units.
-axis.
clockwise about the origin. Draw a sketch to help visualize what this looks like.
Use the following diagram that shows a transformation of
5. What kind of transformation was used to go from
to
to
6. Use the distance formula to show that
7. Is the transformation congruence preserving? Justify your answer.
Use the following diagram for exercises 8-9.
8. What kind of transformation is shown above?
9. Is this transformation congruence preserving? Justify your answer.
262
for exercises 5-7:
?
10. Can a
rotation be described in terms of reflections? Justify your answer.
Answers
1.
2.
3.
4.
5. Reflection about the
-axis
6.
8. This is a dilation.
9. No, we can see that each side of the larger triangle is twice as long as the corresponding side in the
original, so this is not length preserving.
10. Yes, a
rotation about the origin is the same as two reflections done consecutively, one across the
-axis and then one across the
-axis. For a
; rotation, the rule for transforming coordinates is
. Now, suppose point
is reflected twice, first across the
-axis, and then across
263
264
the
-axis. After the first transformation, the coordinates are
the
- axis, we get
Then after reflection on
, which is the same coordinates that result from a
rotation.
5. Relationships Within Triangles
Midsegments of a triangle
Learning Objectives
•
Identify the midsegment of a triangle.
•
Apply the Midsegment Theorem to solve problems involving side lengths and midsegments of triangles.
•
Use the Midsegment Theorem to solve problems involving variable side lengths and midsegments of
triangles.
Introduction
In previous lessons, we used the parallel postulate to learn new theorems that enabled us to solve a variety
of problems about parallel lines:
Parallel Postulate: Given: line
through
that is parallel to .
and a point
not on
. There is exactly one line
In this lesson we extend these results to learn about special line segments within triangles. For example,
the following triangle contains such a configuration:
Triangle
is cut by
where
and
is called a midsegment of
are midpoints of sides
. Note that
and
respectively.
has other midsegments in addition to
. Can you see where they are in the figure above?
If we construct the midpoint of side
at point
the following figure and see that segments
and
and construct
and
are midsegments of
respectively, we have
.
265
In this lesson we will investigate properties of these segments and solve a variety of problems.
Properties of midsegments within triangles
We start with a theorem that we will use to solve problems that involve midsegments of triangles.
Midsegment Theorem: The segment that joins the midpoints of a pair of sides of a triangle
is:
1. parallel to the third side.
2. half as long as the third side.
Proof of 1. We need to show that a midsegment is parallel to the third side. We will do this using the Parallel
Postulate.
Consider the following triangle
. Construct the midpoint
By the Parallel Postulate, there is exactly one line though
intersects side
that
at point
. We will show that
of side
.
that is parallel to side
must be the midpoint of
is a midsegment of the triangle and is parallel to
. Let’s say that it
and then we can conclude
.
We must show that the line through
and parallel to side
will intersect side
at its midpoint. If
a parallel line cuts off congruent segments on one transversal, then it cuts off congruent segments on every
transversal. This ensures that point
Since
of side
, we have
.
is the midpoint of side
. Hence, by the definition of midpoint, point
is a midsegment of the triangle and is also parallel to
Proof of 2. We must show that
In
266
.
, construct the midpoint of side
is the midpoint
.
.
at point
and midsegments
and
as follows:
First note that
by part one of the theorem. Since
and
and
, then
since alternate interior angles are congruent. In addition,
.
Hence,
by The ASA Congruence Postulate.
of congruent triangles are congruent. Since
is the midpoint of
since corresponding parts
, we have
and
by segment addition and substitution.
So,
and
.
Example 1
Use the Midsegment Theorem to solve for the lengths of the midsegments given in the following figure.
,
and
Theorem to find
A.
are midpoints of the sides of the triangle with lengths as indicated. Use the Midsegment
.
B. The perimeter of the triangle
A. Since
.
is a midpoint, we have
.
and
. By the theorem, we must have
267
B. By the Midsegment Theorem,
implies that
; similarly,
, and
Hence, the perimeter is
We can also examine triangles where one or more of the sides are unknown.
Example 2
Use the Midsegment Theorem to find the value of
in the following triangle having lengths as indicated and midsegment
By the Midsegment Theorem we have
.
. Solving for
, we have
Lesson Summary
In this lesson we:
•
Introduced the definition of the midsegment of a triangle and examined examples.
•
Stated and proved the Midsegment Theorem.
•
Solved problems using the Midsegment Theorem.
Lesson Exercises
are midpoints of sides of triangles
and
Complete the following:
1. If
268
, then
2. If
, then
3. If
and
___ and
___.
____.
, then
___and
___.
.
=
.
.
4. If
and
5. Consider triangle
, then
_____.
with vertices
a. Find the coordinates of point
and midpoint
on
.
.
b. Use the Midsegment Theorem to find the coordinates of the point
midsegment.
6. For problem 5, describe another way to find the coordinates of point
Theorem.
on side
that makes
the
that does not use the Midsegment
In problems 7-8, the segments join the midpoints of two sides of the triangle. Find the values of
for each problem.
and
7.
8.
269
9. In triangle
, sides
,
, and
have lengths
is formed by joining the midpoints of
and
. Find the perimeter of
respectively. Triangle
.
10.
a. For the original triangle
of 9, find its perimeter and compare to the perimeter of
.
b. Can you state a relationship between a triangle’s perimeter and the perimeter of the triangle formed by
connecting its midsegments?
11. Given:
Prove:
12. Given:
270
is the midpoint of
,
,
.
,
,
, and
.
is the midpoint of
.
Can you conclude that
If true, prove the assertion. If false, provide a counterexample.
Answers
1.
and
2.
3.
,
4.
5.
a.
b.
6. Find midpoint
and then the slope of
the equation of the line that includes
7.
,
8.
,
. Find the line through
(line
parallel to
). Find the intersection of lines
and
(line
). Find
.
9.
10.
a. The perimeter of
is
The perimeter of
is
b. The perimeter of the midsegment triangle will always be half the perimeter of the original triangle.
11. Use the givens and Theorem 5-1 to show that point
is the midpoint of
.
12. The assertion is true. Using Theorem 5-1, it can be shown that the triangles are congruent by SSS
postulate.
Perpendicular Bisectors in Triangles
Learning Objectives
•
Construct the perpendicular bisector of a line segment.
•
Apply the Perpendicular Bisector Theorem to identify the point of concurrency of the perpendicular bisectors of the sides (the circumcenter).
271
•
Use the Perpendicular Bisector Theorem to solve problems involving the circumcenter of triangles.
Introduction
In our last lesson we examined midsegments of triangles. In this lesson we will examine another construction
that can occur within triangles, called perpendicular bisectors.
The perpendicular bisector of a line segment is the line that:
1. divides the line segment into two congruent sub-segments.
2. intersects the line segment at a right angle.
Here is an example of a perpendicular bisector to line segment
.
Perpendicular Bisector Theorem and its Converse
We can prove the following pair of theorems about perpendicular bisectors.
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment,
then it is equidistant from the endpoints of the segment.
Proof. Consider
with perpendicular bisector
We must show that
"1. Since
So
, it follows that
on line
as follows:
and angles
.
by CPCTC (corresponding parts of congruent triangles are congruent).
It turns out that we can also prove the converse of this theorem.
272
and
.
is the perpendicular bisector of
are congruent and are right angles.
By the SAS postulate, we have
with points
and
Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the
endpoints of a segment, then the point is on the perpendicular bisector of the segment.
as follows with
Proof. Consider
We will construct the midpoint
of
1. Construct the midpoint of
at point
2. Consider
and
3. So by CPCTC, we have
4. Since
and
. Hence,
.
and show that
Construct
is the perpendicular bisector to
.
.
. These are congruent triangles by postulate SSS.
.
form a straight angle and are also congruent, then
is on the perpendicular bisector to
.
Notice that we just proved the Perpendicular Bisector Theorem and we also proved the Converse of the
Perpendicular Bisector Theorem. When you prove a theorem and its converse you have proven a biconditional
statement. We can state the Perpendicular Bisector Theorem and its converse in one step: A point is on the
perpendicular bisector of a segment if and only if that point is equidistant from the endpoints of the segment.
We will now use these theorems to prove an interesting result about the perpendicular bisectors of the sides
of a triangle.
Concurrency of Perpendicular Bisectors: The perpendicular bisectors of the sides of
a triangle intersect in a point that is equidistant from the vertices.
Proof. We will use the previous two theorems to establish the proof.
1. Consider
273
2. We can construct the perpendicular bisectors of sides
and
3. We will show that point
also lies on the perpendicular bisector of
,
and
.
the vertices
4. Construct line segments
5. Since
,
and
is on the perpendicular bisector of
dicular Bisector Theorem and
is equidistant from
and
6. By the transitive law, we have
we must have that
intersecting at point
as follows.
and thus is equidistant from
as follows.
, then
. Similarly,
is equidistant from
and
by the Perpen-
is on the perpendicular bisector of
by the Perpendicular Bisector Theorem. Therefore,
, then
.
. By the Converse of the Perpendicular Bisector Theorem,
is on the perpendicular bisector of
.
The point
has a special property. Since it is equidistant from each vertex, we can see that
is the
center of a circle that circumscribes the triangle. We call
the circumcenter of the triangle. This is illustrated in the following figure.
274
Example 1
Construct a circumscribed triangle using a compass and a straightedge.
1. Draw triangle
with your straightedge.
2. Use your compass to construct the perpendicular bisectors of the sides and find the point of concurrency
.
3. Use your compass to verify that
.
4. Use your compass to construct the circle that circumscribes
275
Example 2
Construct a circumscribed triangle using The Geometer’s Sketchpad (GSP)
We can use the commands of GSP to construct the circumcenter and corresponding circle as follows.
1. Open a new sketch and construct triangle
using the Segment Tool.
2. You can construct the perpendicular bisectors of the sides by going to the Construct menu and choosing
the following options. Select each side and choose Construct Midpoints. Then for each side select the midpoint
and the side (and nothing else), and then choose Construct Perpendicular Bisector.
3. Select two of the three bisectors and choose Construct Point of Intersection from the Construct menu.
This will provide point
, the circumcenter.
4. Construct the circle having center
and passing through points
and
. Recall that there are
two ways to construct the circle: . Using the draw tool on the left column, and . Using the Construct
Menu. For this construction, you will want to use the Construct menu to ensure that the circle passes through
the vertices.
276
As a further exploration, try the following with paper:
1. Cut out any triangle from a sheet of paper.
2. Fold the the triangle over one side so that the side is folded in half.
3. Repeat for the other two sides.
4. What do you notice?
Notice that the folds will cross at the circumcenter, unless the triangle is obtuse. In which case the fold lines
will meet outside the triangle if they continued.
Lesson Summary
In this lesson we:
•
Defined the perpendicular bisector of a line segment.
•
Stated and proved the Perpendicular Bisector Theorem.
•
Solved problems using the Perpendicular Bisector Theorem.
Points to Consider
If we think about three non-collinear points in a plane, we can imagine a triangle that has each point as a
vertex. Locating the circumcenter, we can draw a circle that all three vertices will be on. What does this tell
us about any three non-collinear points in a plane?
There is a unique circle for any three non-collinear points in the same plane.
Finding a circle through any three points will also work in coordinate geometry. You can use the circumcenter
to find the equation of a circle through any three points. In calculus this method is used (together with some
tools that you have probably not learned yet) to precisely describe the curvature of any curve.
Lesson Exercises
Construct the circumcenter of
and the circumscribed circle for each of the following triangles using
a straightedge, compass, and Geometer's Sketchpad.
1.
277
2.
3.
4. Based on your constructions in 1-3, state a conjecture about the relationship between a triangle and the
location of its circumcenter.
5. In this lesson we found that we could circumscribe a triangle by finding the point of concurrency of the
perpendicular bisectors of each side. Use Geometer's Sketchpad to see if the method can be used to circumscribe each of the following figures:
a. a square
278
b. a rectangle
c. a parallelogram
d. From your work in a-c, what condition must hold in order to circumscribe a quadrilateral?
6. Consider equilateral triangle
. Construct the perpendicular bisectors of the sides of the triangle
and the circumcenter
. Connect the circumcenter to each vertex. Your original triangle is now divided
into six triangles. What can you conclude about the six triangles?
279
7. Suppose three cities
and
are situated as follows.
The leaders of these cities wish to construct a new health center that is equidistant from each city. Is this a
wise plan? Why or why not?
8. True or false: An isosceles triangle will always have its circumcenter located inside the triangle? Give
reasons for your answer.
9. True or false: The perpendicular bisectors of an equilateral triangle intersect in the exact center of the
triangle’s interior. Give reasons for your answer.
10. Consider line segment
so that
. Suppose that we wish to find point
is equilateral. How can you use perpendicular bisectors to find the location of point
11. Suppose that
Construct the bisector of
280
with coordinates
is a
right triangle as indicated:
.
?
Prove:
is the perpendicular bisector of
.
Answers
1.
2.
3.
281
4. If triangle
is acute, then the circumcenter lies inside of the triangle. If triangle
then the circumcenter lies outside of the triangle. If triangle
lies on the hypotenuse of the triangle.
is obtuse,
is a right triangle, then the circumcenter
5.
a. Yes
b. Yes
c. No
d. Opposite angles must be supplementary.
6.
a. The triangles are congruent to one another and each is a
right triangle.
7. It is not a wise plan. Since
and
form an obtuse triangle, the location of the circumcenter would
be outside the triangle. Hence, the health center would be located at the circumcenter, which would be a
much greater distance from each city than the distance between the cities themselves.
8. False. It is possible to have isosceles triangles that are acute, obtuse, and right. Hence, there are
isosceles triangles where the circumcenter could be located outside the triangle (in the case of an obtuse
triangle) or on the boundary of the triangle (in the case of a right triangle).
282
9. True. See the solution to problem 10 to verify this fact.
10.
a. Construct the perpendicular bisector of
b. Point
.
. Note that the slope of
is
.
will be located on the perpendicular bisector. The perpendicular bisector will have slope
c. The perpendicular bisector will pass through the midpoint
and have slope
Its equation
is
d. So, the distance from
to
e. Consider the distance from point
dinate of point
.
is equal to
to
f. Since
lies on the line
-coordinate.
, which is
.
. Solve the following distance equation to find the
, use the value of
coor-
found from the equation to find the
use CPCTC and the definitions of bisector and properties about
11. Show that
congruent adjacent angles forming a straight angle.
Angle Bisectors in Triangles
Learning Objectives
•
Construct the bisector of an angle.
•
Apply the Angle Bisector Theorem to identify the point of concurrency of the perpendicular bisectors of
the sides (the incenter).
•
Use the Angle Bisector Theorem to solve problems involving the incenter of triangles.
Introduction
In our last lesson we examined perpendicular bisectors of the sides of triangles. We found that we were
able to use perpendicular bisectors to circumscribe triangles. In this lesson we will learn how to inscribe
circles in triangles. In order to do this, we need to consider the angle bisectors of the triangle. The bisector
of an angle is the ray that divides the angle into two congruent angles.
Here is an example of an angle bisector in an equilateral triangle.
283
Angle Bisector Theorem and its Converse
We can prove the following pair of theorems about angle bisectors.
Angle Bisector Theorem: If a point is on the bisector of an angle, then the point is
equidistant from the sides of the angle.
Before we proceed with the proof, let’s recall the definition of the distance from a point to a line. The distance
from a point to a line is the length of the line segment that passes through the point and is perpendicular to
the original line.
Proof. Consider
side through point
with angle bisector
as follows:
We will show that
.
1. Since
addition, since
is the bisector of
and
, then
and
, perpendicular to each
by the definition of angle bisector. In
are perpendicular to the sides of
right angles and thus congruent. Finally,
284
, and segments
, then
by the reflexive property.
and
are
2. By the AAS postulate, we have
3. So
.
by CPCTC (corresponding parts of congruent triangles are congruent).
Therefore
is equidistant from each side of the angle. And since
represents any point on the angle bisector, we can say that every point on the angle bisector is equidistant from the sides of the angle.
We can also prove the converse of this theorem.
Converse of the Angle Bisector Theorem: If a point is in the interior of an angle and
equidistant from the sides, then it lies on the bisector of the angle.
Proof. Consider
with points
and
and segment
1. As the distance to each side is given by the lengths of
and
are perpendicular to sides
and
is the hypotenuse of right triangles
, and
4-6.
, and
3.
4. Hence, point
and
as follows:
respectively, we have that
respectively.
2. Note that
and
such that
and
Hence, since
are right angles, then the triangles are congruent by Theorem
by CPCTC.
lies on the angle bisector of
.
Notice that we just proved the Angle Bisector Theorem (If a point is on the angle bisector then it is equidistant
from the sides of the angle) and we also proved the converse of the Angle Bisector theorem (If a point is
equidistant from the sides of an angle then it is on the angle bisector of the triangle). When we have proven
both a theorem and its converse we say that we have proven a biconditional statement. We can put the
two conditional statements together using if and only if: "A point is on the angle bisector of an angle if and
only if it is equidistant from the sides of the triangle."
Angle Bisectors in a Triangle
We will now use these theorems to prove an interesting result about the angle bisectors of a triangle.
Concurrency of Angle Bisectors Theorem: The angle bisectors of a triangle intersect
in a point that is equidistant from the three sides of the triangle.
Proof. We will use the previous two theorems to establish the proof.
285
1. Consider
.
2. We can construct the angle bisectors of
3. We will show that point
of
.
and
is equidistant from sides
4. Construct perpendicular line segments from point
5. Since
Therefore,
6. Since
is on the bisectors of
is equidistant from
angle bisector of
,
and
,
, and
to sides
and
is equidistant from sides
intersecting at point
and that
,
, and
as follows.
is on the bisector
as follows:
, then by Theorem 5-5,
, and
.
.
, Theorem 5-6 applies and we must have that
is on the
.
The point
has a special property. Since it is equidistant from each side of the triangle, we can see that
is the center of a circle that lies within the triangle. We say that the circle is inscribed within the triangle
and the point
is called the incenter of the triangle. This is illustrated in the following figure.
Example 1
286
Inscribe the following triangle using a compass and a straightedge.
1. Draw triangle
with your straightedge.
2. Use your compass to construct the angle bisectors and find the point of concurrency
3. Use your compass to construct the circle that inscribes
.
.
Example 2
Inscribe a circle within the following triangle using The Geometer’s Sketchpad.
We can use the commands of GSP to construct the incenter and corresponding circle as follows:
1. Open a new sketch and construct triangle
using the Segment Tool.
287
2. You can construct the angle bisectors of the angles by first designating the angle by selecting the appropriate vertices (e.g., to select the angle at vertex
, select points
,
and
in order) and then
choosing Construct Angle Bisector from the Construct menu. After bisecting two angles, construct the point
of intersection by selecting each angle bisector and choosing Intersection from the Construct menu. (Recall
from our proof of the concurrency of angle bisectors theorem that we only need to bisect two of the angles
to find the incenter.)
3. You are now ready to construct the circle. Recall that the radius of the circle must be the distance from
to each side – our figure above does not include that segment. However, we do not need to construct
the perpendicular line segments as we did to prove Theorem 5-7. Sketchpad will measure the distance for
us.
to each side, select point
and one side of the triangle. Choose
4. To measure the distance from
Distance from the Measure menu. This will give you the radius of your circle.
5. We are now ready to construct the circle. Select point
and the distance from
to the side of the
triangle. Select “Construct circle by center + radius” from the Construct menu. This will give the inscribed
circle within the triangle.
Lesson Summary
In this lesson we:
288
•
Defined the angle bisector of an angle.
•
Stated and proved the Angle Bisector Theorem.
•
Solved problems using the Angle Bisector Theorem.
•
Constructed angle bisectors and the inscribed circle with compass and straightedge, and with Geometer’s
Sketchpad.
Points to Consider
How are circles related to triangles, and how are triangles related to circles? If we draw a circle first, what
are the possibilities for the triangles we can circumscribe? In later chapters we will more carefully define
and work with the properties of circles.
Lesson Exercises
Construct the incenter of
and the inscribed circle for each of the following triangles using a
straightedge, compass, and Geometer's Sketchpad.
3. In the last lesson we found that we could circumscribe some kinds of quadrilaterals as long as opposite
angles were supplementary. Use Geometer's Sketchpad to explore the following quadrilaterals and see if
you can inscribe them by the angle bisector method.
a. a square
289
b. a rectangle
c. a parallelogram
d. a rhombus
290
e. From your work in a-d, what condition must hold in order to circumscribe a quadrilateral?
4. Consider equilateral triangle
. Construct the angle bisectors of the triangle and the incenter
. Connect the incenter to each vertex so that the line segment intersects the side opposite the angle as follows.
As with circumcenters, we get six congruent
the sides. What kind of figure do you get?
triangles. Now connect the points that intersect
5. True or false: An incenter can also be a circumcenter. Illustrate your reasoning with a drawing.
6. Consider the situation described in exercise 4 for the case of an isosceles triangle. What can you conclude
about the six triangles that are formed?
7. Consider line segment
with coordinates
,
. Suppose that we wish to find points
and
so that the resulting quadrilateral can be either circumscribed or inscribed. What are some pos291
sibilities for locating points
and
8. Using a piece of tracing or Patty Paper, construct an equilateral triangle. Bisect one angle by folding one
side onto another. Unfold the paper. What can you conclude about the fold line?
9. Repeat exercise 8 with an isosceles triangle. What can you conclude about repeating the folds?
10. What are some other kinds of polygons where you could use Patty Paper to bisect an angle into congruent
figures?
11.
Given:
is the perpendicular bisector of
is the perpendicular bisector of
Prove:
.
12.
Given:
bisects
.
bisects
Prove:
Answers
1.
292
bisects
.
.
.
.
2.
3.
a. Yes
b. No: Bisectors are not concurrent at a point.
c. No: Bisectors are not concurrent at a point.
d. Yes
e. Angle bisectors must be concurrent.
4. Equilateral triangle
293
5. The statement is true in the case of an equilateral triangle. In addition, for squares the statement is also
true.
6. We do not get six congruent triangles as before. But we get four congruent triangles and a separate pair
of congruent triangles. In addition, if we connect the points where the bisectors intersect the sides, we get
an isosceles triangle.
7. From our previous exercises we saw that we could inscribe and circumscribe some but not all types of
quadrilaterals. Drawing from those exercises, we see that we could circumscribe and inscribe a square. So,
locating the points at
and
and
is one such possibility. Similarly, we could locate the points at
and get a kite that can be inscribed but not circumscribed.
8. The fold line divides the triangle into two congruent triangles and thus is a line of symmetry for the triangle.
Note that the same property will hold by folding at each of the remaining angles.
294
9. There is only one fold line that divides the triangle into congruent triangles, the line that folds the angle
formed by the congruent sides.
10. Any regular polygon will have this property. For example, a regular pentagon:
11.
12.
295
Medians in Triangles
Learning Objectives
•
Construct the medians of a triangle.
•
Apply the Concurrency of Medians Theorem to identify the point of concurrency of the medians of the
triangle (the centroid).
•
Use the Concurrency of Medians Theorem to solve problems involving the centroid of triangles.
Introduction
In our two last lessons we learned to circumscribe circles about triangles by finding the perpendicular bisectors
of the sides and to inscribe circles within triangles by finding the triangle’s angle bisectors. In this lesson we
will learn how to find the location of a point within the triangle that involves the medians.
Definition of Median of a Triangle
A median of a triangle is the line segment that joins a vertex to the midpoint of the opposite side.
Here is an example that shows the medians in an obtuse triangle.
That the three medians appear to intersect in a point is no coincidence. As was true with perpendicular bisectors of the triangle sides and with angle bisectors, the three medians will be concurrent (intersect in a
point). We call this point the centroid of the triangle. We can prove the following theorem about centroids.
The Centroid of a Triangle
Concurrency of Medians Theorem: The medians of a triangle will intersect in a point
that is two-thirds of the distance from the vertices to the midpoint of the opposite side.
Consider
with midpoints of the sides located at
the medians at the centroid,
. The theorem states that
,
, and
,
, and
and the point of concurrency of
.
The theorem can be proved using a coordinate system and the midpoint and distance formulas for line
segments. We will leave the proof to you (Homework Exercise #10), but will provide an outline and helpful
296
hints for developing the proof.
Example 1.
Use The Concurrency of Medians Theorem to find the lengths of the indicated segments in the following
triangle that has medians
,
1. If
, then
____ and
____.
2. If
. then
____ and
_____.
We will start by finding
Now for
, and
as indicated.
.
,
Napoleon's Theorem
In the remainder of the lesson we will provide an interesting application of a theorem attributed to Napoleon
Bonaparte, Emperor of France from 1804 to 1821, which makes use of equilateral triangles and centroids.
We will explore Napoleon’s theorem using The Geometer’s Sketchpad.
But first we need to review how to construct an equilateral triangle using circles. Consider
having equal radius and centered at
and
as follows:
and circles
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Once you have hidden the circles, you will have an equilateral triangle. You can use the construction any
time you need to construct an equilateral triangle by selecting the finished triangle and then making a Tool
using the tool menu.
Preliminary construction for Napoleon’s Theorem: Construct any triangle
equilateral triangle on each side.
298
. Construct an
Find the centroid of each equilateral triangle and connect the centroids to get the Napoleon outer triangle.
Measure the sides of the new triangle using Sketchpad. What can you conclude about the Napoleon outer
triangle? (Answer: The triangle is equilateral.)
This result is all the more remarkable since it applies to any triangle
. You can verify this fact in
GSP by "dragging" a vertex of the original triangle
to form other triangles. The outer triangle will
remain equilateral. Homework problem 9 will allow you to further explore this theorem.
Example 2
Try this:
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1. Draw a triangle on a sheet of card stock paper (or thin cardboard) and locate the centroid.
2. Carefully cut out the triangle.
3. Hold your pencil point up and place the triangle on it so that the centroid rests on the pencil.
4. What do you notice?
The triangle balances on the pencil. Why does the triangle balance?
Lesson Summary
In this lesson we:
•
Defined the centroid of a triangle.
•
Stated and proved the Concurrency of Medians Theorem.
•
Solved problems using the Concurrency of Medians Theorem.
•
Demonstrated Napoleon’s Theorem.
Points to Consider
So far we have been looking at relationships within triangles. In later chapters we will review the area of a
triangle. When we draw the medians of the triangle, six smaller triangles are created. Think about the area
of these triangles, and how that might relate to example 1 above.
Lesson Exercises
1. Find the centroid of
for each of the following triangles using Geometer's Sketchpad. For each
triangle, measure the lengths of the medians and the distances from the centroid to each of the vertices.
What can you conclude for each of the triangles?
a. an equilateral triangle
b. an isosceles triangle
c. A scalene triangle
2.
has points
as midpoints of sides and the centroid located at point
Find the following lengths if
____,
as follows.
___.
3. True or false: A median cannot be an angle bisector. Illustrate your reasoning with a drawing.
4. Find the coordinates of the centroid
300
of
with vertices
, and
.
5. Find the coordinates of the centroid
find
.
of
with vertices
and
. Also,
6. Use the example sketch of Napoleon’s Theorem to form the following triangle:
a. Reflect each of the centroids in the line that is the closest side of the original triangle.
b. Connect the points to form a new triangle that is called the inner Napoleon triangle.
c. What can you conclude about the inner Napoleon triangle?
301
7. You have been asked to design a triangular metal logo for a club at school. Using the following rectangular
coordinates, determine the logo’s centroid.
8. Prove Theorem 5-8. An outline of the proof together with some helpful hints is provided here.
Proof. Consider
with
Hints: Note that the midpoint of side
as follows:
is located at the origin. Construct the median from vertex
. The point of concurrency of the three medians will be located on
the origin, and call it
that is two-thirds of the way from
to the origin.
to
at point
9. Prove: Each median of an equilateral triangle divides the triangle into two congruent triangles.
Answers
1.
a. Medians all have same length; distances from vertices to centroid – all are same; they are two-thirds the
lengths of the medians.
b. Two of the medians have same length; distances from vertices to centroid are same for these two; all
are two-thirds the lengths of the medians.
c. Medians all have different lengths; distances from vertices to centroid; all are different; they are two-thirds
the lengths of the medians.
2.
.
3. False. The statement is true in the case of isosceles (vertex angle) and all angles in an equilateral triangle.
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4.
5.
a.
b.
.
6.
a. The triangle is equilateral.
b. The difference in the areas of the inner and outer triangles is equal to the area of the original triangle.
7. The centroid will be located at
. Note that
from
point to
is located
. The midpoint of the vertical side of the triangle is located at
units from point
and that centroid will be one-third of the distance
.
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8.
a. Note that the midpoint of side
is located at the origin. Construct the median from vertex
origin, and call it
. The point of concurrency of the three medians will be located on
that is two-thirds of the way from
to the origin.
at point
b. Using slopes and properties of straight lines, the point can be determined to have coordinates
,
to the
(
).
c. Use the distance formula to show that point
to the midpoint of the opposite side.
is two-thirds of the way from each of the other two vertices
9. Construct a median in an equilateral triangle. The triangles can be shown to be congruent by the SSS
postulate.
Altitudes in Triangles
Learning Objectives
•
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Construct the altitude of a triangle.
•
Apply the Concurrency of Altitudes Theorem to identify the point of concurrency of the altitudes of the
triangle (the orthocenter).
•
Use the Concurrency of Altitudes Theorem to solve problems involving the orthocenter of triangles.
Introduction
In this lesson we will conclude our discussions about special line segments associated with triangles by
examining altitudes of triangles. We will learn how to find the location of a point within the triangle that involves
the altitudes.
Definition of Altitude of a Triangle
An altitude of a triangle is the line segment from a vertex perpendicular to the opposite side. Here is an
example that shows the altitude from vertex
in an acute triangle.
We need to be careful with altitudes because they do not always lie inside the triangle. For example, if the
triangle is obtuse, then we can easily see how an altitude would lie outside of the triangle. Suppose that we
in the following obtuse triangle:
wished to construct the altitude from vertex
In order to do this, we must extend side
Will the remaining altitudes for
Answer: The altitude from vertex
triangle.
as follows:
(those from vertices
and
) lie inside or outside of the triangle?
will lie inside the triangle; the altitude from vertex
will lie outside the
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As was true with perpendicular bisectors (which intersect at the circumcenter), and angle bisectors (which
intersect at the incenter), and medians (which intersect at the centroid), we can state a theorem about the
altitudes of a triangle.
Concurrency of Triangle Altitudes Theorem: The altitudes of a triangle will intersect
in a point. We call this point the orthocenter of the triangle.
Rather than prove the theorem, we will demonstrate it for the three types of triangles (acute, obtuse, and
right) and then illustrate some applications of the theorem.
Acute Triangles
The orthocenter lies within the triangle.
Obtuse Triangles
The orthocenter lies outside of the triangle.
Right Triangles
The legs of the triangle are altitudes. The orthocenter lies at the vertex of the right angle of the triangle.
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Even with these three cases, we may still encounter special triangles that exhibit interesting properties.
Example 1
Use a piece of Patty Paper (
tracing paper), or any square piece of paper to explore orthocenters of isosceles
. Note: Patty Paper may be purchased in bulk from many Internet sites.
Determine any relationships between the location of the orthocenter and the location of the incenter, circumcenter, and centroid.
First let’s recall that you can construct an isosceles triangle with Patty Paper as follows:
1. Draw line segment
2. Fold point
onto point
.
to find the fold line.
3. Locate point
anywhere on the fold line and connect point
to points
and
. (Hint: Locate point
as far away from
and
as possible so that you end up with a good-sized triangle.). Trace three
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copies of
).
onto Patty Paper (so that you end up with four sheets of paper, each showing
4. For one of the sheets, fold the paper to locate the median, angle bisector, and perpendicular bisector
. What do you observe? (Answer: They are the same line segment.)
relative to the vertex angle at point
Fold to find another bisector and locate the intersection of the two lines, the incenter.
5. For the second sheet, locate the circumcenter of
6. For the third sheet, locate the centroid of
7. For the third sheet, locate the orthocenter of
.
.
.
8. Trace the location of the circumcenter, centroid, and orthocenter onto the original triangle. What do you
observe about the four points? (Answer: The incenter, orthocenter, circumcenter, and centroids are collinear
and lie on the median from the vertex angle.)
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Do you think that the four points will be collinear for all other kinds of triangles? The answer is pretty interesting! In our homework we will construct the four points for a more general case.
Lesson Summary
In this lesson we:
•
Defined the orthocenter of a triangle.
•
Stated the Concurrency of Altitudes Theorem.
•
Solved problems using the Concurrency of Altitudes Theorem.
•
Examined the special case of an isosceles triangle and determined relationships about among the incenter,
circumcenter, centroid, and orthocenter.
Points to Consider
Remember that the altitude of a triangle is also its height and can be used to find the area of the triangle.
The altitude is the shortest distance from a vertex to the opposite side.
Lesson Exercises
1. In our lesson we looked at the special case of an isosceles triangle and determined relationships about
among the incenter, circumcenter, centroid, and orthocenter. Explore the case of an equilateral triangle
and see which (if any) relationships hold.
2. Perform the same exploration for an acute triangle. What can you conclude?
3. Perform the same exploration for an obtuse triangle. What can you conclude?
4. Perform the same exploration for a right triangle. What can you conclude?
5. What can you conclude about the four points for the general case of
?
6. In 3 you found that three of the four points were collinear. The segment joining these three points define
the Euler segment. Replicate the exploration of the general triangle case and measure the lengths of the
Euler segment and the sub-segments. Drag your drawing so that you can investigate potential relationships
for several different triangles. What can you conclude about the lengths?
7. (Found in Exploring Geometry, 1999, Key Curriculum Press ) Construct a triangle and find the Euler
segment. Construct a circle centered at the midpoint of the Euler segment and passing through the midpoint
of one of the sides of the triangle.
This circle is called the nine-point circle. The midpoint it passes through is one of the nine points. What are
the other eight?
8. Consider
with,
,
altitudes of the triangles as indicated.
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Prove:
.
9. Consider isosceles triangle
Prove:
with
, and
,
.
.
Answers
1. All four points are the same.
2. The four points all lie inside the triangle.
3. The four points all lie outside the triangle.
4. The orthocenter lies on the vertex of the right angle and the circumcenter lies on the midpoint of the hypotenuse.
5. The orthocenter, the circumcenter, and the centroid are always collinear.
6.
a. The circumcenter and the orthocenter are the endpoints of the Euler segment.
b. The distance from the orthocenter to the centroid is twice the distance from the centroid to the circumcenter.
7. Three of the points are the midpoints of the triangle’s sides. Three other points are the points where the
altitudes intersect the opposite sides of the triangle. The last three points are the midpoints of the segments
connecting the orthocenter with each vertex.
8. The congruence can be proven by showing the congruence of triangles
can be done by applying postulate AAS to the two triangles.
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and
. This
9. The proof can be completed by using the AAS postulate to show that triangles
are congruent. The conclusion follows from CPCTC.
and
Inequalities in Triangles
Learning Objectives
•
Determine relationships among the angles and sides of a triangle.
•
Apply the Triangle Inequality Theorem to solve problems.
Introduction
In this lesson we will examine the various relationships among the measure of the angles and the lengths
of the sides of triangles. We will do so by stating and proving a few key theorems that will enable us to determine the types of relationships that hold true.
Look at the following two triangles
We see that the first triangle is isosceles while in the the second triangle,
are the measures of the angles at
and
related to the lengths of
fact it is the case) that the measure of the angle at vertex
is larger than
is longer than
and
. How
It appears (and, in
.
In this section we will formally prove theorems that reveal when such relationships hold. We will start with
the following theorem.
Relationship Between the Sides and Angles of a Triangle
Theorem: If two sides of a triangle are of unequal length, then the angles opposite these
sides are also unequal. The larger side will have a larger angle opposite it.
Proof. Consider
with
1. By the Ruler Postulate, there is a point
angles , and as follows.
. We must show that
on
such that
.
. Construct
and label
311
2. Since
is isosceles, we have
.
3. By angle addition, we have
4. So
5. Note that
.
, and by substitution
is exterior to
6. Hence,
.
, so
.
and
, we have
.
We can also prove a similar theorem about angles.
Larger angle has longer opposite side: If one angle of a triangle has greater measure
than a second angle, then the side opposite the first angle is longer than the side opposite
the second angle.
Proof. In order to prove the theorem, we will use a method that relies on indirect reasoning, a method that
we will explore further. The method relies on starting with the assumption that the conclusion of the theorem
is wrong, and then reaching a conclusion that logically contradicts the given statements.
1. Consider
with
2. Assume temporarily that
3. If
statements.
312
. We must show that
is not greater than
, then the angles at vertices
and
. Then either
.
or
.
are congruent. This is a contradiction of our given
4. If
, then
by the fact that the longer side is opposite the largest
angle (the theorem we just proved). But this too contradicts our given statements. Hence, we must have
.
With these theorems we can now prove an interesting corollary.
Corollary The perpendicular segment from a point to a line is the shortest segment from
the point to the line.
Proof. The proof is routine now that we have proved the major results.
Consider point
and line
and the perpendicular line segment from
We can draw the segment from
to any point on line
to
as follows.
and get the case of a right triangle as follows:
Since the triangle is a right triangle, the side opposite the right angle (the hypotenuse) will always have
to , which is opposite an angle of
length greater than the length of the perpendicular segment from
90°.
Now we are ready to prove one of the most useful facts in geometry, the triangle inequality theorem.
Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
Proof. Consider
. We must show the following:
1.
2.
3.
313
Suppose that
is the longest side. Then statements 2 and 3 above are true.
In order to prove 1.
side as follows:
, construct the perpendicular from point
to
on the opposite
Now we have two right triangles and can draw the following conclusions:
is
Since the perpendicular segment is the shortest path from a point to a line (or segment), we have
the shortest segment from
and
to
. Also,
is the shortest segment from
to
. Therefore
and by addition we have
.
So,
.
Example 1
Can you have a triangle with sides having lengths
?
Without a drawing we can still answer this question—it is an impossible situation, we cannot have such a
triangle. By the Triangle Inequality Theorem, we must have that the sum of lengths of any two sides of the
triangle must be greater than the length of the third side. In this case, we note that
Example 2
Find the angle of smallest measure in the following triangle.
314
.
has the smallest measure. Since the triangle is a right triangle, we can find
Pythagorean Theorem (which we will prove later).
using the
By the fact that the longest side is opposite the largest angle in a triangle, we can conclude that
.
Lesson Summary
In this lesson we:
•
Stated and proved theorems that helped us determine relationships among the angles and sides of a
triangle.
•
Introduced the method of indirect proof.
•
Applied the Triangle Inequality Theorem to solve problems.
Points to Consider
Knowing these theorems and the relationships between the angles and sides of triangles will be applied
when we use trigonometry. Since the size of the angle affects the length of the opposite side, we can show
that there are specific angles associated with certain relationships (ratios) between the sides in a right triangle,
and vice versa.
Lesson Exercises
1. Name the largest and smallest angles in the following triangles:
a.
b.
315
2. Name the longest side and the shortest side of the triangles.
a.
b.
3. Is it possible to have triangles with the following lengths? Give a reason for your answer.
a.
b.
316
c.
d.
4. Two sides of a triangle have lengths
side?
and
5. The base of an isosceles triangle has length
. What can you conclude about the length of the third
. What can you say about the length of each leg?
In exercises 6 and 7, find the numbered angle that has the largest measure of the triangle.
6.
7.
In exercises 8-9, find the longest segment in the diagram.
8.
317
9.
10.
Given:
Prove:
11.
318
,
Given:
Prove:
Answers
1.
a.
is largest and
is smallest.
b.
is largest and
is smallest.
a.
is longest and
is the shortest.
b.
is longest and
is the shortest.
2.
3.
a. No,
.
b. Yes
c. No,
.
d. Yes
4. The third side must have length
such that
5. The legs each must have length greater than
.
.
6.
7.
8.
9.
10. Since the angle opposite each of the two segments that comprise
the corresponding segments of
,
is greater than the angle opposite
.
11. Proof
319
Inequalities in Two Triangles
Learning Objectives
•
Determine relationships among the angles and sides of two triangles.
•
Apply the SAS and SSS Triangle Inequality Theorems to solve problems.
Introduction
In our last lesson we examined the various relationships among the measure of angles and the lengths of
the sides of triangles and proved the Triangle Inequality Theorem that states that the sum of the lengths of
two sides of a triangle is greater than the third side. In this lesson we will look at relationships in two triangles.
SAS Inequality Theorem
Let’s begin our discussion by looking at the following congruent triangles.
If we think of the sides of the triangle as matchsticks that are "hinged" at
and
respectively then we
can increase the measure of the angles by opening up the sticks. If we open them so that
, then we see that
. Conversely, if you open them so that
. We can prove theorems that involve these relationships.
, then we see that
SAS Inequality Theorem (The Hinge Theorem): If two sides of a triangle are congruent
to two sides of another triangle, but the included angle of the first triangle has greater
measure than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
Proof. Consider
and
must show that
Construct
is on
320
,
,
, take point
so that
. We
.
so that
or
with
is not on
=
. On
. In either case, we must have
. Either
by SAS postulate and
by CPCTC.
Case 1:
is on
By the Segment Addition Postulate
, so
. By substitution we have
we had
Case 2:
is not on
. But from our congruence above
and we have proven case 1.
.
so that it intersects
Construct the bisector of
Recall that
.
Note that
by SAS postulate. Then
at point
So,
by the Triangle Inequality Theorem.
Now
by the segment addition postulate, and
of
, so by substitution we have
. Draw
and
.
.
by our original construction
or
and we have proven
case 2.
We can also prove the converse of the Hinge theorem.
SSS Inequality Theorem-Converse of Hinge Theorem: If two sides of a triangle are
congruent to two sides of another triangle, but the third side of the first triangle is longer
than the third side of the second triangle, then the included angle of the first triangle is
greater in measure than the included angle of the second triangle.
Proof. In order to prove the theorem, we will again use indirect reasoning as we did in proving Theorem 511.
321
Consider
and
with
,
,
We must show that
.
1. Assume that
is not greater than
. Then either
or
.
Case 1: If
have
, then
and
are congruent by SAS postulate and we
. But this contradicts the given condition that
, then
2. Case 2: If
condition that
.
by Theorem 5-11. This contradicts the given
.
3. Since we get contradictions in both cases, then our original assumption was incorrect and we must have
.
We can now look at some problems that we could solve with these theorems.
Example 1
What we can deduce from the following diagrams.
1. Given:
322
is a median of
Since
and
2. Given:
as indicated.
with
.
then Theorem 5-14 applies and we have
.
Since we have two sides of
and we have
congruent with two sides of
, then Theorem 5-13 applies
.
Lesson Summary
In this lesson we:
•
Stated and proved theorems that helped determine relationships among the angles and sides of a pair
of triangle.
•
Applied the SAS and SSS Inequality Theorems to solve problems.
Lesson Exercises
Use the theorems to make deductions in problems 1-5. List any theorems or postulates you use.
1.
2. Suppose that
is acute and
is obtuse
3.
323
4.
5.
In problems 6-10, determine whether the assertion is true and give reasons to support your answers.
324
6. Assertion:
and
7. Assertion:
in the figure below.
.
8. Assertion:
.
In problems 9-10, is the assertion true or false?
9. Assertion:
10. Consider
Assertion:
.
is a right triangle with median from
as indicated.
.
325
Answers
1.
by Theorem 5-13.
2.
by Theorem 5-13.
3.
by Theorem 5-14.
4. We cannot deduce anything as we know nothing about the included angle nor the third side of each triangle.
5.
by Theorem 5-13.
6. The assertions are true.
by Theorem 5-14. Since both triangles are isosceles and
, then an implication of the fact that base angles are congruent will imply that
7. The assertion is false. The two triangles have two sides congruent. The measure of angle
to the
angle) is
.
(adjacent
since the triangle is equilateral. Hence, Theorem 5-13 applies and so
8. The assertion is true by Theorem 5-14:
9. The assertion is false. Theorem 5-14 applies and we have
.
.
.
10. The assertion is false. We do not have enough information to apply the theorems in this example.
Indirect Proof
Learning Objective
•
Reason indirectly to develop proofs of statement
Introduction
Recall that in proving Theorems about the relationship between the sides and angles of triangle we used a
method of proof in which we temporarily assumed that the conclusions was false and then reached a contradiction of the given statements. This method of proving something is called indirect proof. In this lesson
we will practice using indirect proofs with both algebraic and geometric examples.
Indirect Proofs in Algebra
Let’s begin our discussion with an algebraic example that we will put into if-then form.
326
Example 1
If
, then
Proof. Let’s assume temporarily that the
. Then we can reach a contradiction by applying
our standard algebraic properties of real numbers and equations as follows:
This last statement contradicts the given statement that
we must have
Hence, our assumption is incorrect and
.
We can also employ this kind of reasoning in geometric situations. Consider the following theorem which
we have previously proven using the Corresponding Angles Postulate:
Theorem: If parallel lines are cut by a transversal, then alternate interior angles are
congruent.
Proof. It suffices to prove the theorem for one pair of alternate interior angles. So consider
We need to show that
.
and
.
Assume that we have parallel lines and that
. We know that lines are parallel, so we have
by postulate
that corresponding angles are congruent and
. Since vertical angles are
congruent, we have
. So by substitution, we must have
, which is a contradiction.
Lesson Summary
In this lesson we:
327
•
Illustrated some examples of proof by indirect reasoning, from algebra and geometry.
Points to Consider
Indirect reasoning can be a powerful tool in proofs. In the section on logical reasoning we saw that if there
are two possibilities for a statement (such as TRUE or FALSE), if we can show one of them is not true (i.e.
show that a statement is NOT FALSE), then the opposite possibility is all we have left (i.e. the statement is
TRUE).
Lesson Exercises
Generate a proof by contradiction for each of the following statements.
1. If
is an integer and
2. If in
is even, then
we have
3. If
, then
is even.
, then
is not equilateral.
.
4. If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are
parallel.
5. If one angle of a triangle is larger than another angle of a triangle, then the side opposite the larger angle
is longer than the side opposite the smaller angle.
6. The base angles of an isosceles triangle are congruent.
7. If
is an integer and
8. If we have
is odd, then
with
=
is odd.
, then
is not a right angle.
9. If two angles of a triangle are not congruent, then the sides opposite those angles are not congruent.
10. Consider the triangle the following figure with
.
bisect
, and
. Prove that
does not
Answers
1. Assume
is odd. Then
contradicts the given statement that
328
for some integer
is even.
, and
which is odd. This
2. Assume
is equilateral. Then by definition, the sides are congruent. By the parallel postulate,
we can construct a line parallel to the base through point
as follows:
From this we can show with alternate interior angles that the triangle is equiangular so
. This contradicts the given statement that
3. Assume that
or
is not greater than
. Then either
in which case
, which is a contradiction,
, also a contradiction
.
4. Assume that the lines are not parallel. Then
we have
=
.
, in which case we can solve the quadratic inequality to get
of the fact that
=
. This is a contradiction of the fact that
. But
=
for vertical angles, so
.
5. Hint: This is theorem 5-11.
6. Assume
, say
. By Theorem 5-11 we have
contradicts the fact that we have an isosceles triangle.
, which
329
7. Proof follows the lesson example closely. Assume
even, which is a contradiction.
8. In
we have
substitution we have
is even. Then it can be shown that
. Assume that
so that
must be
is a right angle. Hence, by
, which is a contradiction.
9. Suppose we have a triangle in which the sides opposite two angles are congruent. Then it follows that
the triangle must be isosceles. By Isosceles Triangle Theorem, the opposite angles are congruent, which
is a contradiction.
does bisect
10. Assume
by CPCTC, which is a contradiction.
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. Hence, the two triangles are congruent by SAS and
6. Quadrilaterals
Interior Angles
Learning Objectives
•
Identify the interior angles of convex polygons.
•
Find the sums of interior angles in convex polygons.
•
Identify the special properties of interior angles in convex quadrilaterals.
Introduction
By this point, you have studied the basics of geometry and you’ve spent some time working with triangles.
Now you will begin to see some ways to apply your geometric knowledge to other polygons. This chapter
focuses on quadrilaterals—polygons with four sides.
Note: Throughout this chapter, any time we talk about polygons, we will assume that we are talking about
convex polygons.
Interior Angles in Convex Polygons
The interior angles are the angles on the inside of a polygon.
As you can see in the image, a polygon has the same number of interior angles as it does sides.
Summing Interior Angles in Convex Polygons
You have already learned the Triangle Sum Theorem. It states that the sum of the measures of the interior
angles in a triangle will always be
. What about other polygons? Do they have a similar rule?
We can use the triangle sum theorem to find the sum of the measures of the angles for any polygon. The
first step is to cut the polygon into triangles by drawing diagonals from one vertex. When doing this you must
make sure none of the triangles overlap.
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Notice that the hexagon above is divided into four triangles.
Since each triangle has internal angles that sum to
, you can find out the sum of the interior angles
in the hexagon. The measure of each angle in the hexagon is a sum of angles from the triangles. Since
none of the triangles overlap, we can obtain the TOTAL measure of interior angles in the hexagon by summing
all of the triangles' interior angles. Or, multiply the number of triangles by
:
The sum of the interior angles in the hexagon is
.
Example 1
What is the sum of the interior angles in the polygon below?
The shape in the diagram is an octagon. Draw triangles on the interior using the same process.
The octagon can be divided into six triangles. So, the sum of the internal angles will be equal to the sum
of the angles in the six triangles.
So, the sum of the interior angles is
.
What you may have noticed from these examples is that for any polygon, the number of triangles you can
draw will be two less than the number of sides (or the number of vertices). So, you can create an expression
for the sum of the interior angles of any polygon using
for the number of sides on the polygon.
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The
sum
of
the
interior
angles
of
a
polygon
with
sides
is
.
Example 2
What is the sum of the interior angles of a nonagon?
To find the sum of the interior angles in a nonagon, use the expression above. Remember that a nonagon
will be equal to nine.
has nine sides, so
So, the sum of the interior angles in a nonagon is
.
Interior Angles in Quadrilaterals
A quadrilateral is a polygon with four sides, so you can find out the sum of the interior angles of a convex
quadrilateral using our formula.
Example 3
What is the sum of the interior angles in a quadrilateral?
Use the expression to find the value of the interior angles in a quadrilateral. Since a quadrilateral has four
sides, the value of
will be 4.
So, the sum of the measures of the interior angles in a quadrilateral is
.
This will be true for any type of convex quadrilateral. You’ll explore more types later in this chapter, but they
will all have interior angles that sum to
. Similarly, you can divide any quadrilateral into two triangles.
This will be helpful for many different types of proofs as well.
Lesson Summary
In this lesson, we explored interior angles in polygons. Specifically, we have learned:
•
How to identify the interior angles of convex polygons.
•
How to find the sums of interior angles in convex polygons.
•
How to identify the special properties of interior angles in convex quadrilaterals.
Understanding the angles formed on the inside of polygons is one of the first steps to understanding shapes
and figures. Think about how you can apply what you have learned to different problems as you approach
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them.
Lesson Exercises
1. Copy the polygon below and show how it can be divided into triangles from one vertex.
2. Using the triangle sum theorem, what is the sum of the interior angles in this pentagon?
3-4: Find the sum of the interior angles of each polygon below.
3.
Number of sides =
Sum of interior angles =
4.
Number of sides =
Sum of interior angles =
5. Complete the following table:
Polygon name
Number of sides
triangle
4
5
6
7
octagon
decagon
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Sum of measures of interior angles
6. A regular polygon is a polygon with congruent sides and congruent angles. What is the measure of each
angle in a regular pentagon?
7. What is the measure of each angle in a regular octagon?
8. Can you generalize your answer from 6 and 7? What is the measure of each angle in a regular
-gon?
9. Can you use the polygon angle sum theorem on a convex polygon? Why or why not? Use the convex
quadrilateral
to explain your answer.
10. If we know the sum of the angles in a polygon is
the work leading to your answer.
, how many sides does the polygon have? Show
Answers
1. One possible answer: [Diff: 1]
2.
[Diff: 1]
3. Number of sides
, sum of interior angles =
[Diff: 1]
4. Number of sides
, sum of interior angles =
[Diff: 2]
5. [Diff: 2-3]
Polygon name
Number of sides
triangle
3
quadrilateral
4
pentagon
5
hexagon
6
heptagon
7
octagon
8
Sum of measures of interior angles
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decagon
10
dodecagon
12
-gon
6. Since the sum of the angles is
7.
8.
, each angle measures
[Diff: 2]
[Diff: 2]
[Diff: 3]
9. Answers will vary. One possibility is no, we cannot use the polygon angle sum theorem because
is
an acute angle that does not open inside the polygon. Alternatively, if we allow for angles between
and
, then we can use the angle sum theorem, but so far we have not seen angles measuring more
[Diff: 3].
than
10. Solve the equation: [Diff: 3]
Exterior Angles
Learning Objectives
•
Identify the exterior angles of convex polygons.
•
Find the sums of exterior angles in convex polygons.
Introduction
This lesson focuses on the exterior angles in a polygon. There is a surprising feature of the sum of the exterior
angles in a polygon that will help you solve problems about regular polygons.
Exterior Angles in Convex Polygons
Recall that interior means inside and that exterior means outside. So, an exterior angle is an angle on the
outside of a polygon. An exterior angle is formed by extending a side of the polygon.
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As you can tell, there are two possible exterior angles for any given vertex on a polygon. In the figure above
we only showed one set of exterior angles; the other set would be formed by extending each side in the
opposite (clockwise) direction. However, it doesn’t matter which exterior angles you use because on each
vertex their measurement will be the same. Let’s look closely at one vertex, and draw both of the exterior
angles that are possible.
As you can see, the two exterior angles at the same vertex are vertical angles. Since vertical angles are
congruent, the two exterior angles possible around a single vertex are congruent.
Additionally, because the exterior angle will be a linear pair with its adjacent interior angle, it will always be
supplementary to that interior angle. As a reminder, supplementary angles have a sum of
.
Example 1
What is the measure of the exterior angle
in the diagram below?
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The interior angle is labeled as
. Since you need to find the exterior angle, notice that the interior angle
and the exterior angle form a linear pair. Therefore the two angles are supplementary—they sum to
. So, to find the measure of the exterior angle, subtract
from
.
The measure of
is
.
Summing Exterior Angles in Convex Polygons
By now you might expect that if you add up various angles in polygons, there will be some sort of pattern
or rule. For example, you know that the sum of the interior angles of a triangle will always be
. From
that fact, you have learned that you can find the sums of the interior angles of any polygons with
sides
using the expression
looking at a triangle.
. There is also a rule for exterior angles in a polygon. Let’s begin by
To find the exterior angles at each vertex, extend the segments and find angles supplementary to the interior
angles.
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The sum of these three exterior angles is:
So, the exterior angles in this triangle will sum to
.
To compare, examine the exterior angles of a rectangle.
In a rectangle, each interior angle measures
. Since exterior angles are supplementary to interior angles,
all exterior angles in a rectangle will also measure
.
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Find the sum of the four exterior angles in a rectangle.
So, the sum of the exterior angles in a rectangle is also
.
In fact, the sum of the exterior angles in any convex polygon will always be
many sides the polygon has, the sum will always be
.
. It doesn’t matter how
We can prove this using algebra as well as the facts that at any vertex the sum of the interior and one of
the exterior angles is always
, and the sum of all interior angles in a polygon is
.
Exterior Angle Sum: The sum of the exterior angles of any convex polygon is
Proof. At any vertex of a polygon the exterior angle and the interior angle sum to 180°. So summing all of
the exterior angles and interior angles gives a total of 180 degrees times the number of vertices:
.
On the other hand, we already saw that the sum of the interior angles was:
.
Putting these together we have
Example 2
What is
in the diagram below?
in the diagram is marked as an exterior angle. So, we need to find the measure of one exterior
angle on a polygon given the measures of all of the others. We know that the sum of the exterior angles on
a polygon must be equal to
, regardless of how many sides the shape has. So, we can set up an
340
equation where we set all of the exterior angles shown (including
. Using subtraction, we can find the value of
.
The measure of the missing exterior angle is
) summed and equal to
.
We can verify that our answer is reasonable by inspecting the diagram and checking whether the angle in
question is acute, right, or obtuse. Since the angle should be obtuse,
is a reasonable answer (assuming
the diagram is accurate).
Lesson Summary
In this lesson, we explored exterior angles in polygons. Specifically, we have learned:
•
How to identify the exterior angles of convex polygons.
•
How to find the sums of exterior angles in convex polygons.
We have also shown one example of how knowing the sum of the exterior angles can help you find the
measure of particular exterior angles.
Lesson Exercises
For exercises 1-3, find the measure of each of the labeled angles in the diagram.
1.
______,
________
2.
______,
______,
_______,
______
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3.
______,
______
4. Draw an equilateral triangle with one set of exterior angles highlighted. What is the measure of each exterior angle? What is the sum of the measures of the three exterior angles in an equilateral triangle?
5. Recall that a regular polygon is a polygon with congruent sides and congruent angles. What is the measure
of each interior angle in a regular octagon?
6. How can you use your answer to 5 to find the measure of each exterior angle in a regular octagon? Draw
a sketch to justify your answer.
7. Use your answer to 6 to find the sum of the measures of the exterior angles of an octagon.
8. Complete the following table assuming each polygon is a regular polygon. Note: This is similar to a previous
exercise with more columns—you can use your answer to that question to help you with this one.
Regular Polygon Number of sides Sum of measures Measure of each Measure of each Sum of measures
name
of interior angles interior angle
exterior angle
of exterior angles
triangle
4
5
6
7
octagon
decagon
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9. Each exterior angle forms a linear pair with its adjacent internal angle. In a regular polygon, you can use
two different formulas to find the measure of each exterior angle. One way is to compute
of each interior angle)... in symbols
(measure
.
Alternatively, you can use the fact that all
exterior angles in an
of each exterior angle with by dividing the sum by
-gon sum to
and find the measure
. Again, in symbols this is
Use algebra to show these two expressions are equivalent.
Answers
1.
,
[Diff: 1]
2.
[Diff: 1]
3.
[Diff: 1]
4. Below is a sample sketch.
Each exterior angle measures
5. Sum of the angles is
, the sum of the three exterior angles is
. So, each angle measures
[Diff: 2].
[Diff: 2].
6. Since each exterior angle forms a linear pair with its adjacent interior angle, we can find the measure of
each exterior angle with
7.
[Diff: 2].
[Diff: 2]
8. [Diff: 3]
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Regular Poly- Number Sum of mea- Measure of each interior Measure of each exterior Sum of measures of exgon name
of sides sures of inte- angle
angle
terior angles
rior angles
triangle
3
square
4
pentagon
5
hexagon
6
heptagon
7
octagon
8
decagon
10
dodecagon
12
-gon
9. One possible answer. [Diff: 3]
Classifying Quadrilaterals
Learning Objectives
344
•
Identify and classify a parallelogram.
•
Identify and classify a rhombus.
•
Identify and classify a rectangle.
•
Identify and classify a square.
•
Identify and classify a kite.
•
Identify and classify a trapezoid.
•
Identify and classify an isosceles trapezoid.
•
Collect the classifications in a Venn diagram.
•
Identify how to classify shapes on a coordinate grid.
Introduction
There are many different classifications of quadrilaterals. In this lesson, you will explore what defines each
type of quadrilateral and also what properties each type of quadrilateral has. You have probably heard of
many of these shapes before, but here we will focus on things we’ve learned about other polygons—the
relationships among interior angles, and the relationships among the sides and diagonals. These issues
will be explored in later lessons to further your understanding.
Parallelograms
A parallelogram is a quadrilateral with two pairs of parallel sides. Each of the shapes shown below is a
parallelogram.
As you can see, parallelograms come in a variety of shapes. The only defining feature is that opposite
sides are parallel. But, once we know that a figure is a parallelogram, we have two very useful theorems
we can use to solve problems involving parallelograms: the Opposite Sides Theorem and the Opposite
Angles Theorem.
We prove both of these theorems by adding an auxiliary line and showing that a parallelogram can be divided
into two congruent triangles. Then we apply the definition of congruent triangles—the fact that if two triangles
are congruent, all their corresponding parts are congruent (CPCTC).
An auxiliary line is a line that is added to a figure without changing the given information. You can always
add an auxiliary line to a figure by connecting two points because of the Line Postulate. In many of the proofs
in this chapter we use auxiliary lines.
Opposite Sides of Parallelogram Theorem: The opposite sides of a parallelogram are
congruent.
Proof.
•
Given Parallelogram ABCD
•
Prove
and
Statement
1.
is a parallelogram.
Reason
1. Given
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2. Draw Auxiliary segment
the angles as follows.
and label 2. Line Postulate
3. Definition of parallelogram
3.
4. Alternate Interior Angles Theorem
4.
5. Definition of parallelogram
5.
6.
6. Alternate Interior Angles Theorem
7.
7. Reflexive Property
8. ASA Triangle Congruence Postulate
8.
9.
9. Definition of congruent triangles (all corresponding sides and angles are congruent)
and
Opposite Angles in Parallelogram Theorem: The opposite angles of a parallelogram
are congruent.
Proof. This proof is nearly the same as the one above and you will do it as an exercise.
Rhombi
A rhombus (plural is rhombi or rhombuses) is a quadrilateral that has four congruent sides. While it is
possible for a rhombus to have four congruent angles, it’s only one example. Many rhombi do NOT have
four congruent angles.
Theorem: A rhombus is a parallelogram
Proof.
•
•
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Given: Rhombus
Prove:
and
Statement
1.
Reason
1. Given
is a Rhombus.
2. Definition of a rhombus
2.
3. Add auxiliary segment
4.
5.
6.
7.
.
3. Line Postulate
4. Reflexive Property
5. SSS
6. Definition of Congruent Triangles
7. Converse of AIA Theorem
8.
8. Definition of Congruent Triangles
9.
9. Converse of AIA Theorem
That may seem like a lot of work just to prove that a rhombus is a parallelogram. But, now that you know
that a rhombus is a type of parallelogram, then you also know that the rhombus inherits all of the properties
of a parallelogram. This means if you know something is true about parallelograms, it must also be true
about a rhombus.
Rectangle
A rectangle is a quadrilateral with four congruent angles. Since you know that any quadrilateral will have
interior angles that sum to
interior angle.
(using the expression
), you can find the measure of each
Rectangles will have four right angles, or four angles that are each equal to
.
Square
A square is both a rhombus and a rectangle. A square has four congruent sides as well as four congruent
angles. Each of the shapes shown below is a square.
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Kite
A kite is a different type of quadrilateral. It does not have parallel sides or right angles. Instead, a kite is
defined as a quadrilateral that has two distinct pairs of adjacent congruent sides. Unlike parallelograms or
other quadrilaterals, the congruent sides are adjacent (next to each other), not opposite.
Trapezoid
A trapezoid is a quadrilateral that has exactly one pair of parallel sides. Unlike the parallelogram that has
two pairs, the trapezoid only has one. It may or may not contain right angles, so the angles are not a distinguishing characteristic. Remember that parallelograms cannot be classified as trapezoids. A trapezoid is
classified as having exactly one pair of parallel sides.
Isosceles Trapezoid
An isosceles trapezoid is a special type of trapezoid. Like an isosceles triangle, it has two sides that are
congruent. As a trapezoid can only have one pair of parallel sides, the parallel sides cannot be congruent
(because this would create two sets of parallel sides). Instead, the non-parallel sides of a trapezoid must
be congruent.
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Example 1
Which is the most specific classification for the figure shown below?
A. parallelogram
B. rhombus
C. rectangle
D. square
The shape above has two sets of parallel sides, so it is a parallelogram. It also has four congruent sides,
making it a rhombus. The angles are not right angles (and we can’t assume we know the angle measures
since they are unmarked), so it cannot be a rectangle or a square. While the shape is a parallelogram, the
most specific classification is rhombus. The answer is choice B.
Example 2
Which is the most specific classification for the figure shown below? You may assume the diagram is drawn
to scale.
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A. parallelogram
B. kite
C. trapezoid
D. isosceles trapezoid
The shape above has exactly one pair of parallel sides, so you can rule out parallelogram and kite as possible classifications. The shape is definitely a trapezoid because of the one pair of parallel sides. For a shape
to be an isosceles trapezoid, the other sides must be congruent. That is not the case in this diagram, so the
most specific classification is trapezoid. The answer is choice C.
Using a Venn Diagram for Classification
You have just explored many different rules and classifications for quadrilaterals. There are different ways
to collect and understand this information, but one of the best methods is to use a Venn Diagram. Venn
Diagrams are a way to classify objects according to their properties. Think of a rectangle. A rectangle is a
type of parallelogram (you can prove this using the Converse of the Interior Angles on the Same Side of the
Transversal Theorem), but not all parallelograms are rectangles. Here’s a simple Venn Diagram of that relationship:
Notice that all rectangles are parallelograms, but not all parallelograms are rectangles. If an item falls into
more than one category, it is placed in the overlapping section between the appropriate classifications. If it
does not meet any criteria for the category, it is placed outside of the circles.
To begin organizing the information for a Venn diagram, you can analyze the quadrilaterals we have discussed
thus far by three characteristics: parallel sides, congruent sides, and congruent angles. Below is a table that
shows how each quadrilateral fits these characteristics.
Shape
Number of pairs of parallel Number of pairs of congruent Four congruent angles
sides
sides
Parallelogram
2
2
No
Rhombus
2
2
No
Rectangle
2
2
Yes
Square
2
2
Yes
Kite
0
2
No
Trapezoid
1
0
No
Isosceles trapezoid 1
1
No
Example 3
Organize the classification information in the table above in a Venn Diagram.
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To begin a Venn Diagram, you must first draw a large ellipse representing the biggest category. In this
case, that will be quadrilaterals.
Now, one class of quadrilaterals are parallelograms—all quadrilaterals with opposite sides parallel. But, not
all quadrilaterals are parallelograms: kites have no pairs of parallel sides, and trapezoids only have one pair
of parallel sides. In the diagram we can show this as follows:
Okay, we are almost there, but there are several types of parallelograms. Squares, rectangles, and rhombi
are all types of parallelograms. Also, under the category of trapezoids we need to add isosceles trapezoids.
The completed Venn diagram is like this:
You can use this Venn Diagram to quickly answer questions. For instance, is every square a rectangle?
(Yes.) Is every rhombus a square? (No, but some are.)
Strategies for Shapes on a Coordinate Grid
You have already practiced some of the tricks for analyzing shapes on a coordinate grid. You actually have
all of the tools you need to classify any quadrilateral placed on a grid. To find out whether sides are congruent,
you can use the distance formula.
Distance
Formula:
Distance
between
points
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To find out whether lines are parallel, you can find the slope by computing
. If
the slopes are the same, the lines are parallel. Similarly, if you want to find out if angles are right angles,
you can test the slopes of their lines. Perpendicular lines will have slopes that are opposite reciprocals of
each other.
Example 4
Classify the shape on the coordinate grid below.
First identify whether the sides are congruent. You can use the distance formula four times to find the distance
between the vertices.
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For segment
, find the distance between (-1,3) and (1,9).
For segment
, find the distance between (1, 9) and (3, 3).
For segment
, find the distance between (3, 3) and (1, -3).
For segment
, find the distance between (-1, 3) and (1, -3).
So, the length of each segment is equal to
, and the sides are all equal. At this point, you know that
the figure is either a rhombus or a square. To distinguish, you’ll have to identify whether the angles are right
angles. If one of the angles is a right angle, they all must be, so the shape will be a square. If it isn’t a right
angle, then none of them are, and it is a rhombus.
You can check whether two segments form a right angle by finding the slopes of two intersecting segments.
If the slopes are opposite reciprocals, then the lines are perpendicular and form right angles.
The slope of segment
can be calculated by finding the “rise over the run”.
Now find the slope of an adjoining segment, like
The two slopes are -3 and 3. These are opposite numbers, but they are not reciprocals. Remember that
the opposite reciprocal of -3 would be
, so segments
and
are not perpendicular. Since the
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sides of
do not intersect a right angle, you can rule out square. Therefore
is a rhombus.
Lesson Summary
In this lesson, we explored quadrilateral classifications. Specifically, we have learned:
•
How to identify and classify a parallelogram.
•
How to identify and classify a rhombus.
•
How to identify and classify a rectangle.
•
How to identify and classify a square.
•
How to identify and classify a kite.
•
How to identify and classify a trapezoid.
•
How to identify and classify an isosceles trapezoid.
•
How to collect the classifications in a Venn diagram.
•
How to identify and classify shapes using a coordinate grid.
It is important to be able to classify different types of quadrilaterals in many different situations. The more
you understand the differences and similarities between the shapes, the more success you’ll have applying
them to more complicated problems.
Lesson Exercises
354
1.
____,
_____
2.
____,
_____
3.
____,
_____
Use the diagram below for exercises 4-7:
4. Find the slope of
and
, and find the slope of
and
5. Based on 4, what can you conclude now about quadrilateral
6. Find
7. If
using the distance formula. What can you conclude about
, find
and
.
?
?
.
8. Prove the Opposite Angles Theorem: The opposite angles of a parallelogram are congruent.
9. Draw a Venn diagram representing the relationship between Widgets, Wookies, and Wooblies (these are
made-up terms) based on the following four statements:
a. All Wookies are Wooblies
b. All Widgets are Wooblies
c. All Wookies are Widgets
d. Some Widgets are not Wookies
10. Sketch a kite. Describe the symmetry of the kite and write a sentence about what you know based on
the symmetry of a kite.
Answers
1.
[Diff: 1]
2.
[Diff: 1]
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3.
[Diff: 1]
4. The slope of
and the slope of
both = 0 since the lines are horizontal. For
. Finally for
,
,
[Diff: 2].
5. Since the slopes of the opposite sides are equal, the opposite sides are parallel. Therefore,
a parallelogram [Diff: 2].
is
6. Using the distance formula,
Since
7.
is a parallelogram, we know that
and
[Diff: 2].
[Diff: 2]
8. First, we convert the theorem into “given” information and what we need to prove: Given: Parallelogram
.
Prove:
and
Statement
Reason
1. ABCD is a parallelogram
1. Given
2. Draw auxiliary segment
follows
3.
4.
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and label the angles as 2. Line Postulate
3. Definition of parallelogram
4. Alternate Interior Angles Theorem
5. Definition of parallelogram
5.
6.
6. Alternate Interior Angles Theorem
7.
7. Reflexive property
8. ASA Triangle Congruence Postulate
8.
9.
9. Definition of congruent triangles (all corresponding sides and angles of congruent triangles are congruent)
10.
10. Angle addition postulate
11.
11. Angle addition postulate
12.
12. Substitution
Now we have shown that opposite angles of a parallelogram are congruent [Diff: 3].
9. [Diff: 3]
10. See below. The red line is a line of reflection. Given this symmetry, we can conclude that
[Diff: 3].
Using Parallelograms
Learning Objectives
•
Describe the relationships between opposite sides in a parallelogram.
•
Describe the relationship between opposite angles in a parallelogram.
•
Describe the relationship between consecutive angles in a parallelogram.
357
•
Describe the relationship between the two diagonals in a parallelogram.
Introduction
Now that you have studied the different types of quadrilaterals and their defining characteristics, you can
examine each one of them in greater depth. The first shape you’ll look at more closely is the parallelogram.
It is defined as a quadrilateral with two pairs of parallel sides, but there are many more characteristics that
make a parallelogram unique.
Opposite Sides in a Parallelogram
By now, you recognize that there are many types of parallelograms. They can look like squares, rectangles,
or diamonds. Either way, opposite sides are always parallel. One of the most important things to know,
however, is that opposite sides in a parallelogram are also congruent.
To test this theory, you can use pieces of string on your desk. Place two pieces of string that are the same
length down so that they are parallel. You’ll notice that the only way to connect the remaining vertices will
be two parallel, congruent segments. There will be only one possible fit given two lengths.
Try this again with two pieces of string that are different lengths. Again, lay them down so that they are
parallel on your desk. What you should notice is that if the two segments are different lengths, the missing
segments (if they connect the vertices) will not be parallel. Therefore, it will not create a parallelogram. In
fact, there is no way to construct a parallelogram if opposite sides aren’t congruent.
So, even though parallelograms are defined by their parallel opposite sides, one of their properties is that
opposite sides be congruent.
Example 1
Parallelogram
is shown on the following coordinate grid. Use the distance formula to show that
opposite sides in the parallelogram are congruent.
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You can use the distance formula to find the length of each segment. You are trying to prove that
the same as
, and that
the length of
.)
Start with
So
Next find
So
Next find
is the same as
. The coordinates of
(Recall that
means the same as
are (-4,5) and the coordinates of
are (3,3).
are (3, 3) and the coordinates of
are (6,-4).
are (6,-4) and the coordinates of
are (-1,-2).
is
, or
.
. The coordinates of
.
. The coordinates of
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So
.
Finally, find the length of
So
. The coordinates of
are (-4,5) and the coordinates of
are (-1,-2).
.
Thus, in parallelogram
,
and
. The opposite sides are congruent.
This example shows that in this parallelogram, the opposite sides are congruent. In the last section we
proved this fact is true for all parallelograms using congruent triangles. Here we have shown an example
of this property in the coordinate plane.
Opposite Angles in a Parallelogram
Not only are opposite sides in a parallelogram congruent. Opposite angles are also congruent. You can
prove this by drawing in a diagonal and showing ASA congruence between the two triangles created. Remember that when you have congruent triangles, all corresponding parts will be congruent.
Example 2
Fill in the blanks in the two-column proof below.
•
Given:
•
Prove:
is a parallelogram
Statement
1.
360
is a parallelogram
Reason
1. Given
2. Definition of a parallelogram
2.
3.
____
___
4. ______
_____
3. Alternate Interior Angles Theorem
4. Definition of a parallelogram
5.
5. _____________________
6.
6. Reflexive Property
7.
______
7. ASA Triangle Congruence Postulate
_____
8. Corresponding parts of congruent triangles are congruent
8.
The missing statement in step 3 should be related to the information in step 2.
and
are parallel,
and
is a transversal. Look at the following figure (with the other segments removed) to see the angles
formed by these segments:
Therefore the missing step is
.
Work backwards to fill in step 4. Since step 5 is about
. So step 4 is
, the sides we need parallel are
and
.
The missing reason on step 5 will be the same as the missing reason in step 3: alternate interior angles.
Finally, to fill in the triangle congruence statement, BE CAREFUL to make sure you match up corresponding
angles. The correct form is
bad if you take a few times to get it correct!)
. (Students commonly get this reversed, so don’t feel
As you can imagine, the same process could be repeated with diagonal
to show that
. Opposite angles in a parallelogram are congruent. Or, even better, we can use the
fact that
and
together with the Angle Addition Postulate to show
. We leave the details of these operations for you to fill in.
Consecutive Angles in a Parallelogram
So at this point, you understand the relationships between opposite sides and opposite angles in parallelograms. Think about the relationship between consecutive angles in a parallelogram. You have studied this
scenario before, but you can apply what you have learned to parallelograms. Examine the parallelogram
below.
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Imagine that you are trying to find the relationship between
and
. To help you understand
the relationship, extend all of the segments involved with these angles and remove
.
What you should notice is that
and
are two parallel lines cut by transversal
. So, you can
find the relationships between the angles as you learned in Chapter 1. Earlier in this course, you learned
that in this scenario, two consecutive interior angles are supplementary; they sum to
. The same is
true within the parallelogram. Any two consecutive angles inside a parallelogram are supplementary.
Example 3
Fill in the remaining values for the angles in parallelogram
You already know that
gruent, you can conclude that
below.
since it is given in the diagram. Since opposite angles are con.
Now that you know that consecutive angles are supplementary, you can find the measures of the remaining
angles by subtracting
from
.
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So,
. Since opposite angles are congruent,
will also measure
.
Diagonals in a Parallelogram
There is one more relationship to examine within parallelograms. When you draw the two diagonals inside
parallelograms, they bisect each other. This can be very useful information for examining larger shapes that
may include parallelograms. The easiest way to demonstrate this property is through congruent triangles,
similarly to how we proved opposite angles congruent earlier in the lesson.
Example 4
Use a two-column proof for the theorem below.
•
Given:
•
Prove:
is a parallelogram
and
Statement
1.
is a parallelogram
Reason
1. Given.
3.
2. Opposite sides in a parallelogram are
congruent.
3. Vertical angles are congruent.
4.
4. Alternate interior angles are congruent.
2.
5.
6.
and
5. AAS congruence theorem: If two angles
and one side in a triangle are congruent, the
triangles are congruent.
6. Corresponding parts of congruent triangles are congruent.
Lesson Summary
In this lesson, we explored parallelograms. Specifically, we have learned:
•
How to describe and prove the distance relationships between opposite sides in a parallelogram.
•
How to describe and prove the relationship between opposite angles in a parallelogram.
•
How to describe and prove the relationship between consecutive angles in a parallelogram.
•
How to describe and prove the relationship between the two diagonals in a parallelogram.
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It is helpful to be able to understand the unique properties of parallelograms. You will be able to use this
information in many different ways.
Points to Consider
Now that you have learned the many relationships in parallelograms, it is time to learn how you can prove
that shapes are parallelograms.
Lesson Exercises
1.
2.
______,
_____,
_______,
________
_____
Use the following figure for exercises 3-6.
3. Find the slopes of
and
.
4. Find the slopes of
and
.
5. What kind of quadrilateral is
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? Give an answer that is as detailed as possible.
6. If you add diagonals to
, where will they intersect?
Use the figure below for questions 7-11. Polygon
measurement.
is a regular polygon. Find each indicated
7.
8.
9.
10. What kind of triangle is
?
and add auxiliary lines to make each of the following:
11. Copy polygon
a. a parallelogram
b. a trapezoid
c. an isosceles triangle
Answers
1.
[Diff: 1]
2.
[Diff: 1]
3. Slopes of
and
both = 1 [Diff: 1]
4. Both = -1 [Diff: 2]
5. This figure is a parallelogram since opposite sides have equal slopes (i.e., opposite sides are parallel).
o
Additionally, it is a rectangle because each angle is a 90 angle. We know this because the slopes of adjacent
sides are opposite reciprocals [Diff: 2].
6. The diagonals would intersect at (0,0). One way to see this is to use the symmetry of the figure—each
o
corner is a 90 rotation around the origin from adjacent corners [Diff: 3].
7.
[Diff: 2]
8.
[Diff: 3]
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9.
10.
[Diff: 3]
is an isosceles right triangle [Diff: 3].
11. There are many possible answers. Here is one: Auxiliary lines are in red:
a.
b.
c.
is a parallelogram (in fact it is a rectangle) [Diff: 3].
is a trapezoid [Diff: 3].
is an isosceles triangle [Diff: 3].
Proving Quadrilaterals are Parallelograms
Learning Objectives
•
Prove a quadrilateral is a parallelogram given congruent opposite sides.
•
Prove a quadrilateral is a parallelogram given congruent opposite angles.
•
Prove a quadrilateral is a parallelogram given that the diagonals bisect each other.
•
Prove a quadrilateral is a parallelogram if one pair of sides is both congruent and parallel.
Introduction
You’ll remember from earlier in this course that you have studied converse statements. A converse statement
reverses the order of the hypothesis and conclusion in an if-then statement, and is only sometimes true. For
example, consider the statement: “If you study hard, then you will get good grades.” Hopefully this is true!
However, the converse is “If you get good grades, then you study hard.” This may be true, but is it not necessarily true—maybe there are many other reasons why you get good grades—i.e., the class is really easy!
An example of a statement that is true and whose converse is also true is as follows: If I face east and then
turn a quarter-turn to the right, I am facing south. Similarly, if I turn a quarter-turn to the right and I am facing
south, then I was facing east to begin with.
Also all geometric definitions have true converses. For example, if a polygon is a quadrilateral then it has
four sides and if a polygon has four sides then it is a quadrilateral.
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Converse statements are important in geometry. It is crucial to know which theorems have true converses.
In the case of parallelograms, almost all of the theorems you have studied this far have true converses. This
lesson explores which characteristics of quadrilaterals ensure that they are parallelograms.
Proving a Quadrilateral is a Parallelogram Given Congruent Sides
In the last lesson, you learned that a parallelogram has congruent opposite sides. We proved this earlier
and then looked at one example of this using the distance formula on a coordinate grid to verify that opposite
sides of a parallelogram had identical lengths.
Here, we will show on the coordinate grid that the converse of this statement is also true: If a quadrilateral
has two pairs of opposite sides that are congruent, then it is a parallelogram.
Example 1
Show that the figure on the grid below is a parallelogram.
We can see that the lengths of opposite sides in this quadrilateral are congruent. For example, to find the
length of
we can find the difference in the
-coordinates (6-1 = 5) because
is horizontal (it’s
generally very easy to find the length of horizontal and vertical segments).
and
. So, we have established that opposite sides of this quadrilateral are congruent.
But is it a parallelogram? Yes. One way to argue that CDEF is a parallelogram is to note that
. We can think of
as a transversal that crosses
and
. Now,
interior angles on the same side of the transversal are supplementary, so we can apply the postulate if interior
angles on the same side of the transversal are supplementary then the lines crossed by the transversal are
parallel.
Note: This example does not prove that if opposite sides of a quadrilateral are congruent then the quadrilateral
is a parallelogram. To do that you need to use any quadrilateral with congruent opposite sides, and then
you use congruent triangles to help you. We will let you do that as an exercise, but here’s the basic picture.
What triangle congruence postulate can you use to show
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Proving a Quadrilateral is a Parallelogram Given Congruent Opposite Angles
Much like the converse statements you studied about opposite side lengths, if you can prove that opposite
angles in a quadrilateral are congruent, the figure is a parallelogram.
Example 2
Complete the two-column proof below.
•
Given: Quadrilateral
•
Prove:
with
and
is a parallelogram
Statement
Reason
is a quadrilateral with 1. Given
1.
and
4.
5.
6.
7.
8.
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2
. 2. Sum of the angles in a quadrilateral is
3
.
3.
Substitution
and
4. Combine like terms
5. Factoring
6. Division property of equality (divided both
sides by 2)
7. If interior angles on the same side of a
transversal are supplementary then the lines
crossed by the transversal are parallel
8. Substitution on line 6
9. Same reason as step 7
9.
10.
is a parallelogram
10. Definition of a parallelogram
Proving a Quadrilateral is a Parallelogram Given Bisecting Diagonals
In the last lesson, you learned that in a parallelogram, the diagonals bisect each other. This can also be
turned around into a converse statement. If you have a quadrilateral in which the diagonals bisect each
other, then the figure is a parallelogram. See if you can follow the proof below which shows how this is explained.
Example 3
Complete the two-column proof below.
•
Given:
•
Prove:
, and
is a parallelogram
Statement
Reason
1. Given
1.
2. Given
2.
3. Vertical angles are congruent
3.
4.
If two sides and the angle
between them are congruent, the two triangles are congruent
5. Corresponding parts of congruent triangles are congruent
6. Vertical angles are congruent
4.
5.
6.
7.
8.
9.
is a parallelogram
7.
If two sides and the angle
between them are congruent, then the two
triangles are congruent
8. Corresponding parts of congruent triangle
are congruent
9. If two pairs of opposite sides of a quadrilateral are congruent, the figure is a parallelogram
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So, given only the information that the diagonals bisect each other, you can prove that the shape is a parallelogram.
Proving a Quadrilateral is a Parallelogram Given One Pair of Congruent and Parallel
Sides
The last way you can prove a shape is a parallelogram involves only one pair of sides.
The proof is very similar to the previous proofs you have done in this section so we will leave it as an exercise
for you to fill in. To set up the proof (which often IS the most difficult step), draw the following:
•
•
Given: Quadrilateral ABCD with
Prove:
and
is a parallelogram
Example 4
Examine the quadrilateral on the coordinate grid below. Can you show that it is a parallelogram?
To show that this shape is a parallelogram, you could find all of the lengths and compare opposite sides.
However, you can also study one pair of sides. If they are both congruent and parallel, then the shape is a
parallelogram.
Begin by showing two sides are congruent. You can use the distance formula to do this.
Find the length of
370
. Use
for
and
for
.
Next, find the length of the opposite side,
. Use
for
and
for
.
So,
; they have equal lengths. Now you need to show that
and
are parallel.
You can do this by finding their slopes. Recall that if two lines have the same slope, they are parallel.
So, the slope of
. Now, check the slope of
.
So, the slope of
. Since the slopes of
and
are the same, the two segments are
parallel. Now that have shown that the opposite segments are both parallel and congruent, you can identify
that the shape is a parallelogram.
Lesson Summary
In this lesson, we explored parallelograms. Specifically, we have learned:
371
•
How to prove a quadrilateral is a parallelogram given congruent opposite sides.
•
How to prove a quadrilateral is a parallelogram given congruent opposite angles.
•
How to prove a quadrilateral is a parallelogram given that the diagonals bisect each other.
•
How to prove a quadrilateral is a parallelogram if one pair of sides is both congruent and parallel.
It is helpful to be able to prove that certain quadrilaterals are parallelograms. You will be able to use this
information in many different ways.
Lesson Exercises
Use the following diagram for exercises 1-3.
1. Find each angle:
a.
b.
c.
d.
2. If
and
, find each length:
a.
b.
3. If
and
, find each length:
a.
b.
Use the following figure for exercises 4-7.
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4. Suppose that
,
, and
Give two possible locations for the fourth vertex,
5. Depending on where you choose to put point
Sketch a picture to show why.
6. If you know the parallelogram is named
are three of four vertices (corners) of a parallelogram.
, if you know that the
-coordinate of
is .
in 4, the name of the parallelogram you draw will change.
, what is the slope of the side parallel to
7. Again, assuming the parallelogram is named
, what is the length of
?
?
8. Prove: If opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Given:
with
Prove:
and
and
(i.e.,
is a parallelogram).
9. Prove: If a quadrilateral has one pair of congruent parallel sides, then it is a parallelogram.
10. Note in 9 that the parallel sides must also be the congruent sides for that theorem to work. Sketch a
counterexample to show that if a quadrilateral has one pair of parallel sides and one pair of congruent sides
(which are not the parallel sides) then the resulting figure it is not necessarily a parallelogram. What kind of
quadrilaterals can you make with this arrangement?
Answers
1. a.
, b.
, c.
, d.
need to find almost all angle measures in the diagram to answer this question) [Diff: 1-2].
2. a.
[Diff: 1]
3. a.
4.
5. If
is at
(note: you
, b.
can be at either
is at
or
the parallelogram would be named
then the parallelogram will take the name
(in red in the following illustration). If
.
373
6.
would have a slope of -2.
7.
8. Given:
Prove:
with
and
and
(i.e.,
Statement
is a parallelogram)
1.
Reason
1. Given
2.
2. Given
3. Add auxiliary line
3. Line Postulate
4.
4. Reflexive Property
5. SSS Congruence Postulate
5.
6. Definition of congruent triangles
6.
8.
7. Converse of Alternate Interior Angles
Postulate
8. Definition of congruent triangles
9.
9. Converse of Alternate Interior Angles
Postulate
7.
9. First, translate the theorem into given and prove statements:
Given:
374
with
and
Prove:
Statement
1.
Reason
1. Given
2. Given
2.
3.
3. Alternate Interior Angles Theorem
4. Add auxiliary line
4. Line Postulate
5.
5. Reflexive Property
6. SAS Triangle Congruence Postulate
6.
7.
7. Definition of congruent triangles
8.
8. Converse of Alternate Interior Angles
Theorem
10. If the congruent sides are not the parallel sides, then you can make either a parallelogram (in black) or
an isosceles trapezoid (in red):
Rhombi, Rectangles, and Squares
Learning Objectives
•
Identify the relationship between the diagonals in a rectangle.
•
Identify the relationship between diagonals in a rhombus.
•
Identify the relationship between diagonals and opposite angles in a rhombus.
•
Identify and explain biconditional statements.
Introduction
Now that you have a much better understanding of parallelograms, you can begin to look more carefully
into certain types of parallelograms. This lesson explores two very important types of parallelograms—rectangles and rhombi. Remember that all of the rules that apply to parallelograms still apply to rectangles and
375
rhombi. In this lesson, you’ll learn about rules specific to these shapes that are not true for all parallelograms.
Diagonals in a Rectangle
Recall from previous lessons that the diagonals in a parallelogram bisect each other. You can prove this
with congruence of triangles within the parallelogram. In a rectangle, there is an even more special relationship
between the diagonals. The two diagonals in a rectangle will always be congruent. We can show this using
the distance formula on a coordinate grid.
Example 1
Use the distance formula to demonstrate that the two diagonals in the rectangle below are congruent.
To solve this problem, you need to find the lengths of both diagonals in the rectangle. First, draw line segments that connect the vertices of the rectangle. So, draw a segment from
to
.
You can use the distance formula to find the length of the diagonals. Diagonal
to
goes from
.
Next, find the length of diagonal
376
to
. That diagonal goes from
to
.
and from
So,
. In this example, the diagonals are congruent. Are the diagonals of rectangles
always congruent? The answer is yes.
Theorem: The diagonals of a rectangle are congruent
The proof of this theorem relies on the definition of a rectangle (a quadrilateral in which all angles are congruent) as well as the property that rectangles are parallelograms.
•
Given: Rectangle
•
Prove:
Statement
1.
is a rectangle
Reason
1. Given
2.
2. Definition of a rectangle
3.
3. Opposite sides of a parallelogram are
4.
4. Reflexive Property of
5.
6.
5. SAS Congruence Postulate
6. Definition of congruent triangles (corresponding parts of congruent triangles are congruent)
Perpendicular Diagonals in Rhombi
Remember that rhombi are quadrilaterals that have four congruent sides. They don’t necessarily have right
angles (like squares), but they are also parallelograms. Also, all squares are parallelograms.
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The diagonals of a rhombus not only bisect each other (because they are parallelograms), they do so at a
right angle. In other words, the diagonals are perpendicular. This can be very helpful when you need to
measure angles inside rhombi or squares.
Theorem: The diagonals of a rhombus are perpendicular bisectors of each other
The proof of this theorem uses the fact that the diagonals of a parallelogram bisect each other and that if
two angles are congruent and supplementary, then they are right angles.
•
Given: Rhombus
•
Prove:
with diagonals
Statement
and
intersecting at point A
1.
is a rhombus
Reason
1. Given
2.
is a parallelogram
2. Theorem: All rhombi are parallelograms
3.
3. Definition of a rhombus
4.
4. Reflexive Property of
5.
5. Diagonals of a parallelogram bisect each
other
6. SSS Triangle Congruence Postulate
6.
7.
8.
tary
and
9.
and
10.
7. Definition of congruent triangles (corresponding parts of congruent triangles are
congruent)
are supplemen- 8. Linear Pair Postulate
are right angles 9. Congruent supplementary angles are
right angles
10. Definition of perpendicular lines
Remember that you can also show that lines or segments are perpendicular by comparing their slopes.
Perpendicular lines have slopes that are opposite reciprocals of each other.
Example 2
Analyze the slope of the diagonals in the rhombus below. Use slope to demonstrate that they are perpendicular.
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Notice that the diagonals in this diagram have already been drawn in for you. To find the slope, find the
change in
over the change in
. This is also referred to as rise over run.
Begin by finding the slope of the diagonal
Now find the slope of the diagonal
, which goes from W(-3,2) to Y(5,-2).
from Z(0,-2) to X(2,2).
The slope of
and the slope of
other, so the two segments are perpendicular.
. These two slopes are opposite reciprocals of each
Diagonals as Angle Bisectors
Since a rhombus is a parallelogram, opposite angles are congruent. One property unique to rhombi is that
in any rhombus, the diagonals will bisect the interior angles. Here we will prove this theorem using a different
method than the proof we showed above.
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Theorem: The diagonals of a rhombus bisect the interior angles
Example 3
Complete the two-column proof below.
•
Given:
•
Prove:
is a rhombus
Statement
1.
is a rhombus
Reason
1. Given
2. All sides in a rhombus are congruent
2.
5.
3. Any triangle with two congruent sides is
isosceles
4. The base angles in an isosceles triangle
are congruent
5. Alternate interior angles are congruent
6.
6. Transitive Property
3.
is isosceles
4.
Segment
bisects
. You could write a similar proof for every angle in the rhombus. Diagonals
in rhombi bisect the interior angles.
Biconditional Statements
Recall that a conditional statement is a statement in the form “If ... then ... .” For example, if a quadrilateral
is a parallelogram, then opposite sides are congruent.
You have learned a number of theorems as conditional statements. Many times you have also investigated
the converses of these theorems. Sometimes the converse of a statement is true, and sometimes the converse
are not. For example, you could say that if you live in Los Angeles, you live in California. However, the
converse of this statement is not true. If you live in California, you don’t necessarily live in Los Angeles.
A biconditional statement is a conditional statement that also has a true converse. For example, a true
biconditional statement is, “If a quadrilateral is a square then it has exactly four congruent sides and four
congruent angles.” This statement is true, as is its converse: “If a quadrilateral has exactly four congruent
sides and four congruent angles, then that quadrilateral is a square.” When a conditional statement can be
written as a biconditional, then we use the term “if and only if.” In the previous example, we could say: “A
quadrilateral is a square if and only if it has four congruent sides and four congruent angles.”
Example 4
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Which of the following is a true biconditional statement?
A. A polygon is a square if and only if it has four right angles.
B. A polygon is a rhombus if and only if its diagonals are perpendicular bisectors.
C. A polygon is a parallelogram if and only if its diagonals bisect the interior angles.
D. A polygon is a rectangle if and only if its diagonals bisect each other.
Examine each of the statements to see if it is true. Begin with choice A. It is true that if a polygon is a square,
it has four right angles. However, the converse statement is not necessarily true. A rectangle also has four
right angles, and a rectangle is not necessarily a square. Providing an example that shows something is
not true is called a counterexample.
The second statement seems correct. It is true that rhombi have diagonals that are perpendicular bisectors.
The same is also true in converse—if a figure has perpendicular bisectors as diagonals, it is a rhombus.
Check the other statements to make sure that they are not biconditionally true.
The third statement isn’t necessarily true. While rhombi have diagonals that bisect the interior angles, it is
not true of all parallelograms. Choice C is not biconditionally true.
The fourth statement is also not necessarily true. The diagonals in a rectangle do bisect each other, but
parallelograms that are not rectangles also have bisecting diagonals. Choice D is not correct.
So, after analyzing each statement carefully, only B is true. Choice B is the correct answer.
Lesson Summary
In this lesson, we explored rhombi, rectangles, and squares. Specifically, we have learned:
•
How to identify and prove the relationship between the diagonals in a rectangle.
•
How to identify and prove the relationship between diagonals in a rhombus.
•
How to identify and prove the relationship between diagonals and opposite angles in a rhombus.
•
How to identify and explain biconditional statements.
It is helpful to be able to identify specific properties in quadrilaterals. You will be able to use this information
in many different ways.
Lesson Exercises
Use Rectangle
for exercises 1-3.
381
1. a.
, b.
2. a.
, b.
3. a.
, b.
Use rhombus
for exercises 4-7.
4. If
a.
___
b.
___
5.
a.
b.
382
in. and
in., then
6. What is the perimeter of
7.
?
is the ______________________ of
For exercises 8 and 9, rewrite each given statement as a biconditional statement. Then state whether it is
true. If the statement is false, draw a counterexample.
8. If a quadrilateral is a square, then it is a rhombus.
9. If a quadrilateral has for right angles, then it is a rectangle.
10. Give an example of an if-then statement whose converse is true. Then write that statement as a biconditional.
Answers
1. a.
b.
[Diff: 1]
2. a.
b.
[Diff 1]
3. a.
4. a.
b.
[Diff: 1]
, b.
[Diff: 1]
, [Diff: 2]
5. a.
6. The perimeter is 216 in. [Diff: 2]
7. Perpendicular bisector [Diff: 2]
8. A quadrilateral is a square if and only if it is a rhombus. This is FALSE because some rhombi are not
squares. Quadrilateral
below is a counterexample—it is a rhombus, but not a square [Diff: 3].
9. A quadrilateral has four right angles if and only if it is a rectangle. This is TRUE by the definition of rectangle.
10. Answers will vary, but any geometric definition can be written as a biconditional.
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Trapezoids
Learning Objectives
•
Understand and prove that the base angles of isosceles trapezoids are congruent.
•
Understand and prove that if base angles in a trapezoid are congruent, it is an isosceles trapezoid.
•
Understand and prove that the diagonals in an isosceles trapezoid are congruent.
•
Understand and prove that if the diagonals in a trapezoid are congruent, the trapezoid is isosceles.
•
Identify the median of a trapezoid and use its properties.
Introduction
Trapezoids are particularly unique figures among quadrilaterals. They have exactly one pair of parallel sides
so unlike rhombi, squares, and rectangles, they are not parallelograms. There are special relationships in
trapezoids, particularly in isosceles trapezoids. Remember that isosceles trapezoids have non-parallel sides
that are of the same lengths. They also have symmetry along a line that passes perpendicularly through
both bases.
Isosceles Trapezoid
Non-isosceles Trapezoid
Base Angles in Isosceles Trapezoids
Previously, you learned about the Base Angles Theorem. The theorem states that in an isosceles triangle,
the two base angles (opposite the congruent sides) are congruent. The same property holds true for
isosceles trapezoids. The two angles along the same base in an isosceles triangle will also be congruent.
Thus, this creates two pairs of congruent angles—one pair along each base.
Theorem: The base angles of an isosceles trapezoid are congruent
Example 1
Examine trapezoid
384
below.
What is the measure of angle ADC?
This problem requires two steps to solve. You already know that base angles in an isosceles triangle will
be congruent, but you need to find the relationship between adjacent angles as well. Imagine extending the
parallel segments
and
on the trapezoid and the transversal
labeled
is a consecutive interior angle with
.
. You’ll notice that the angle
Consecutive interior angles along two parallel lines will be supplementary. You can find
subtracting
from
.
So,
measures
. Since
is adjacent to the same base as
trapezoid, the two angles must be congruent. Therefore,
by
in an isosceles
Here is a proof of this property.
•
•
Given: Isosceles trapezoid
with
and
Prove:
385
Statement
1.
Reason
is an isosceles trapezoid with 1. Given
2. Line Postulate
2. Extend
3. Construct
as shown in the figure 3. Parallel Postulate
below such that
with added auxiliary lines and
markings
4.
is a parallelogram
4. Definition of a parallelogram
5.
5. Opposite angles in a parallelogram are
6.
6. Opposite sides of a parallelogram are
congruent
7. Definition of isosceles triangle
7.
is isosceles
8.
8. Base angles in an isosceles triangle are
9.
9. Alternate Interior Angles Theorem
10.
10. Transitive Property of
11.
11. Transitive Property of
Identify Isosceles Trapezoids with Base Angles
In the last lesson, you learned about biconditional statements and converse statements. You just learned
that if a trapezoid is an isosceles trapezoid then base angles are congruent. The converse of this statement
is also true. If a trapezoid has two congruent angles along the same base, then it is an isosceles trapezoid.
You can use this fact to identify lengths in different trapezoids.
First, we prove that this converse is true.
Theorem: If two angles along one base of a trapezoid are congruent, then the trapezoid
is an isosceles trapezoid
•
•
386
Given: Trapezoid
Prove:
with
and
This proof is very similar to the previous proof, and it also relies on isosceles triangle properties.
Statement
1. Trapezoid
has
and
2. Parallel Postulate
2. Construct
3. Corresponding Angles Postulate
3.
4.
is a parallelogram
with auxiliary lines
6. Transitive Property
6.
7.
4. Definition of a parallelogram
5. Opposite sides of a parallelogram are
5.
Trapezoid
Reason
1. Given
is isosceles
7. Definition of isosceles triangle
8.
8. Converse of the Base Angles Theorem
9.
9. Transitive Property
Example 2
What is the length of
in the trapezoid below?
387
Notice that in trapezoid
, two base angles are marked as congruent. So, the trapezoid is
isosceles. That means that the two non-parallel sides have the same length. Since you are looking for the
length of
, it will be congruent to
. So,
feet.
Diagonals in Isosceles Trapezoids
The angles in isosceles trapezoids are important to study. The diagonals, however, are also important. The
diagonals in an isosceles trapezoid will not necessarily be perpendicular as in rhombi and squares. They
are, however, congruent. Any time you find a trapezoid that is isosceles, the two diagonals will be congruent.
Theorem: The diagonals of an isosceles trapezoid are congruent
Example 3
Review the two-column proof below.
•
Given:
•
Prove:
Statement
1.
Reason
1. Given
3.
2. Base angles in an isosceles trapezoid
are congruent
3. Reflexive Property.
4.
4.
5.
5. Corresponding parts of congruent triangles are congruent
2.
388
is a trapezoid and
So, the two diagonals in the isosceles trapezoid are congruent. This will be true in any isosceles trapezoids.
Identifying Isosceles Trapezoids with Diagonals
The converse statement of the theorem stating that diagonals in an isosceles triangle are congruent is also
true. If a trapezoid has congruent diagonals, it is an isosceles trapezoid. You can either use measurements
shown on a diagram or use the distance formula to find the lengths. If you can prove that the diagonals are
congruent, then you can identify the trapezoid as isosceles.
Theorem: If a trapezoid has congruent diagonals, then it is an isosceles trapezoid
Example 4
Is the trapezoid on the following grid isosceles?
It is true that you could find the lengths of the two sides to identify whether or not this trapezoid is isosceles.
However, for the sake of this lesson, compare the lengths of the diagonals.
Begin by finding the length of
.
Now find the length of
. The coordinates of
. The coordinates of
are
and the coordinates of
are (5,5) and the coordinates of
are
are (0,-1).
389
Thus, we have shown that the diagonals are congruent.
is isosceles.
. Therefore, trapezoid
Trapezoid Medians
Trapezoids can also have segments drawn in called medians. The median of a trapezoid is a segment that
connects the midpoints of the non-parallel sides in a trapezoid. The median is located half way between the
bases of a trapezoid.
Example 5
In trapezoid
below, segment
is a median. What is the length of
The median of a trapezoid is a segment that is equidistant between both bases. So, the length of
be equal to half the length of
Therefore,
is 4 inches.
. Since you know that
inches, you can divide that value by 2.
Theorem: The length of the median of a trapezoid is equal to half of the sum of the lengths
of the bases
This theorem can be illustrated in the example above,
390
will
Therefore, the measure of segment
is 7 inches. We leave the proof of this theorem as an exercise,
but it is similar to the proof that the length of the triangle midsegment is half the length of the base of the
triangle.
Lesson Summary
In this lesson, we explored trapezoids. Specifically, we have learned to:
•
Understand and prove that the base angles of isosceles trapezoids are congruent.
•
Understand that if base angles in a trapezoid are congruent, it is an isosceles trapezoid.
•
Understand that the diagonals in an isosceles trapezoid are congruent.
•
Understand that if the diagonals in a trapezoid are congruent, the trapezoid is isosceles.
•
Identify the properties of the median of a trapezoid.
It is helpful to be able to identify specific properties in trapezoids. You will be able to use this information in
many different ways.
Lesson Exercises
Use the following figure for exercises 1-2.
1.
2.
Use the following figure for exercises 3-5.
391
3.
4.
________
5.
Use the following diagram for exercises 6-7.
6.
7.
8. Can the parallel sides of a trapezoid be congruent? Why or why not? Use a sketch to illustrate your answer.
9. Can the diagonals of a trapezoid bisect each other? Why or why not? Use a sketch to illustrate your answer.
10. Prove that the length of the median of a trapezoid is equal to half of the sum of the lengths of the bases.
Answers
1.
2.
3.
4.
5.
392
cm
6.
7.
cm
8. No, if the parallel (and by definition opposite) sides of a quadrilateral are congruent then the quadrilateral
MUST be a parallelogram. When you sketch it, the two other sides must also be parallel and congruent to
each other (proven in a previous section).
9. No, if the diagonals of a trapezoid bisect each other, then you have a parallelogram. We also proved this
in a previous section.
10. We will use a paragraph proof.
Start with trapezoid
and midsegment
.
Now, using the parallel postulate, construct a line through point
intersections as follows:
that is parallel to
. Label the new
393
Now quadrilateral
a parallelogram tells us
is a parallelogram by construction. Thus, the theorem about opposite sides of
The triangle midsegment theorem tells us that
or
So,
by the segment addition postulate
by substitution
by factoring out and canceling the 2
by the segment addition postulate. Which is exactly what we wanted to show!
Kites
Learning Objectives
•
Identify the relationship between diagonals in kites.
•
Identify the relationship between opposite angles in kites.
Introduction
Among all of the quadrilaterals you have studied thus far, kites are probably the most unusual. Kites have
no parallel sides, but they do have congruent sides. Kites are defined by two pairs of congruent sides that
are adjacent to each other, instead of opposite each other.
A vertex angle is between two congruent sides and a non-vertex angle is between sides of different lengths.
Kites have a few special properties that can be proven and analyzed just as the other quadrilaterals you
have studied. This lesson explores those properties.
Diagonals in Kites
The relationship of diagonals in kites is important to understand. The diagonals are not congruent, but they
are always perpendicular. In other words, the diagonals of a kite will always intersect at right angles.
Theorem: The diagonals of a kite are perpendicular
394
This can be examined on a coordinate grid by finding the slope of the diagonals. Perpendicular lines and
segments will have slopes that are opposite reciprocals of each other.
Example 1
Examine the kite
on the following coordinate grid. Show that the diagonals are perpendicular.
To find out whether the diagonals in this diagram are perpendicular, find the slope of each segment and
compare them. The slopes should be opposite reciprocals of each other.
Begin by finding the slope of
change in the
-coordinate.
The slope of
. Remember that the slope is the change in the
is -1. You can also find the slope of
-coordinate over the
using the same method.
395
The slope of
is 1. If you think of both of these numbers as fractions,
and , you can tell that they
are opposite reciprocals of each other. Therefore, the two line segments are perpendicular.
Proving this property in general requires using congruent triangles (surprise!). We will do this proof in two
parts. First, we will prove that one diagonal (connecting the vertex angles) bisects the vertex angles in the
kite.
Part 1:
•
Given: Kite
•
Prove:
with
bisects
and
Statement
1.
and
Reason
1. Given
and
2. Reflexive Property
2.
3. SSS Congruence Postulate
3.
4. Corresponding parts of congruent triangles are congruent
5. Corresponding parts of congruent triangles are congruent
4.
5.
6.
bisects
and
6. Definition of angle bisector
Now we will prove that the diagonals are perpendicular.
Part 2:
396
•
Given: Kite
•
Prove:
with
and
Statement
1. Kite
Reason
and 1. Given
with
2.
2. Reflexive Property of
3.
3. By part 1 above: The diagonal between
vertex angles bisects the angles
4. SAS Congruence Postulate
4.
5.
6.
and
7.
and
5. Corresponding parts of congruent triangles are congruent
are supplementary 6. Linear Pair Postulate
7. Congruent supplementary angles are
are right angles
right angles
8.
8. Definition of perpendicular
Opposite Angles in Kites
In addition to the bisecting property, one other property of kites is that the non-vertex angles are congruent.
So, in the kite PART above,
.
Example 2
Complete the two-column proof below.
•
Given:
•
Prove:
Statement
and
Reason
397
1.
1. Given
2.
3. ___________
4. _____________
2. Given
3. Reflexive Property
4.
If two triangles have three pairs of congruent
sides, the triangles are congruent.
5. ___________________________
5.
We will let you fill in the blanks on your own, but a hint is that this proof is nearly identical to the first proof
in this section.
So, you have successfully proved that the angles between the congruent sides in a kite are congruent.
Lesson Summary
In this lesson, we explored kites. Specifically, we have learned to:
•
Identify the relationship between diagonals in kites.
•
Identify the relationship between opposite angles in kites.
It is helpful to be able to identify specific properties in kites. You will be able to use this information in many
different ways.
Points to Consider
Now that you have learned about different types of quadrilaterals, it is important to learn more about the
relationships between shapes. The next chapter deals with similarity between shapes.
Lesson Exercises
For exercises 1-5, use kite
below with the given measurements.
398
1.
_______
2.
_______
3.
_______
4.
_______
5.
_________
For exercises 6-10, fill in the blanks in each sentence about Kite
below:
6. The vertex angles of kite
are _________ and __________.
7. ___________ is the perpendicular bisector of _______________.
8. Diagonal ___________ bisects
9.
_______
_______,
________ and
_______
_______.
_______, and
_______
_______.
399
10. The line of symmetry in the kite is along segment __________.
11. Can the diagonals of a kite be congruent to each other? Why or why not?
Answers
1.
2.
3.
4.
5.
cm
6. The vertex angles of kite
7.
are
and
is the perpendicular bisector of
8. Diagonal
bisects
and
9. There are many possible answers:
10.
is a line of reflection. Below is kite
fully annotated with geometric markings.
11. No, if the diagonals were congruent then the “kite” would be a square. Since the two pairs of congruent
sides cannot be congruent to each other (i.e., they must be distinct), the diagonals will have different lengths.
400
401
7. Similarity
Ratios and Proportions
Learning Objectives
•
Write and simplify ratios.
•
Formulate proportions.
•
Use ratios and proportions in problem solving.
Introduction
Words can have different meanings, or even shades of meanings. Often the exact meaning depends on the
context in which a word is used. In this chapter you’ll use the word similar.
What does similar mean in ordinary language? Is a rose similar to a tulip? They’re certainly both flowers. Is
an elephant similar to a donkey? They’re both mammals (and symbols of national political parties in the
United States!). Maybe you’d rather say that a sofa is similar to a chair? In loose terms, by similar we usually
mean that things are like each other in some way or ways, but maybe not the same.
Similar has a very precise meaning in geometry, as we’ll see in upcoming lessons. To understand similar
we first need to review some basic skills in ratios and proportions.
Using Ratios
A ratio is a type of fraction. Usually a ratio is a fraction that compares two parts. “The ratio of
be written in several ways.
to
” can
•
•
•
to
Example 1
Look at the data below, giving sales at Bagel Bonanza one day.
Bagel Bonanza Monday Sales
Type of bagel
Number sold
Plain
Cinnamon
Raisin
Sesame
Garlic
Whole grain
403
Everything
a) What is the ratio of the number of cinnamon raisin bagels sold to the number of plain bagels sold?
,
Ratio of cinnamon raisin to plain =
, or
to
.
Note: Depending on the problem, ratios are often written in simplest form. In this case the ratio can be reduced
or simplified because
.
b) What is the ratio, in simplest form, of the number of whole grain bagels sold to the number of
"everything" bagels sold?
Ratio of whole grain to everything =
, or
to
.
c) What is the ratio, in simplest form, of everything bagels sold to the number of whole grain bagels sold?
Answer: This ratio is just the reciprocal of the ratio in b. If the ratio of whole grain to everything is,
,
, or
to
, then the ratio of everything to whole grain is,
,
, or
to
.
d. What is the ratio, in simplest form, of the number of sesame bagels sold to the total number of all bagels
sold?
First find the total number of bagels sold:
.
Ratio of sesame to total sold =
Note that this also means that
,
, or
, or
to
.
, of all the bagels sold were sesame.
In some situations you need to write a ratio of more than two numbers. For example, the ratio, in simplest
form, of the number of cinnamon raisin bagels to the number of sesame bagels to the number of garlic
bagels is
(or
before simplifying).
Example 2
A talent show features only dancers and singers.
•
The ratio of dancers to singers is
•
There are
.
performers in all.
How many singers are there?
There is a whole number
,
404
so that the total number of each group can be represented as
.
Since there are 30 dancers and singers in all,
The number of dancers is
. The number of singers is
these answers. The numbers of dancers and singers have to add up to
. It’s easy to check
, and they have to be in a
ratio.
Check:
. The ratio of dancers to singers is
, or
to
.
Proportions
A proportion is an equation. The two sides of the equation are ratios that are equal to each other. Proportions
are often found in situations involving direct variation. A scale drawing would make a good example.
Example 3
Leo uses a scale drawing of his barn. He recorded actual measurements and the lengths on the scale
drawing that represent those actual measurements.
Barn dimensions
Actual length
Door opening
feet
Interior wall
feet
Water line
feet
Length on scale drawing
inches
inches
?
a) Since he is using a scale drawing, the ratio of actual length to length on the scale drawing should be the
same all the time. We can write two ratios that should be equal. This is the proportion below.
Is the proportion true?
We could write the fractions with a common denominator. One common denominator is
.
.
The proportion is true.
b) Depending on how you think, you might have written a different proportion. You could say that the ratio
of the actual lengths must be the same as the ratio of the lengths on the scale drawing.
.
405
This proportion is also true. One nice thing about working with proportions is that there are several proportions
that correctly represent the same data.
c) What length should Leo use on the scale drawing for the water line?
Let
represent the scale length. Write a proportion.
If two fractions are equal, and they have the same denominator, then the numerators must be equal.
The scale length for the water line is
inches.
Note that the scale for this drawing can be expressed as
inch to
feet, or
inch to
foot.
Proportions and Cross Products
Look at example 3b above.
is true if and only if
.
In the proportion,
,
and are called the means (they’re in the middle);
and
are
called the extremes (they’re on the ends). You can see that for the proportion to be true, the product of the
means
must equal the product of the extremes
. Both products equal
.
It is easy to generalize this means-and-extremes rule for any true proportion.
Means and Extremes Theorem or The Cross Multiplication Theorem
Cross Multiplication Theorem: Let
. If
then
,
,
, and
be real numbers, with
and
.
The proof of the cross multiplication theorem is example 4. The proof of the converse is in the Lesson Exercises.
Example 4
Prove The Cross Multiplication Theorem: For real numbers
, then
406
.
,
,
, and
with
and
If
We will start by summarizing the given information and what we want to prove. Then we will use a twocolumn proof.
•
Given:
•
,
,
, and
are real numbers, with
and
and
Prove:
Statement
1.
,
,
Reason
, and
are real numbers, with 1. Given
and
2. Given
2.
3.
3.
cation
, identity property of multipli-
4. Commutative property of multiplication
4.
5.
5. If equal fractions have the same denominator, then the numerators must be equal
or
This theorem allows you to use the method of cross multiplication with proportions.
Lesson Summary
Ratios are a useful way to compare things. Equal ratios are proportions. With the Means-and-Extremes
Theorem we have a simple but powerful method for solving any proportion.
Points to Consider
Proportions are very “forgiving”—there are many different ways to write proportions that are equivalent to
each other. There are hints of some of these in the Lesson Exercises. In the next lesson, we’ll prove that
these proportions are equivalent.
You know about figures that are congruent. But many figures that are alike are not congruent. They may
have the same shape, even though they are not the same size. These are similar figures; ratios and proportions are integral to defining and understanding similar figures.
Lesson Exercises
The votes for president in a club election were:
Suarez,
Milhone,
Cho,
1. Write each of the following ratios in simplest form.
a. votes for Milhone to votes for Suarez
b. votes for Cho to votes for Milhone
c. votes for Suarez to votes for Milhone to votes for Cho
407
d. votes for Suarez or Cho to total votes
Use the diagram below for exercise 2.
2. Write each of the following ratios in simplest form.
a.
b.
c.
d.
e. area of
area of
f. area of
area of
g. area of
area of
3. The measures of the angles of a triangle are in the ratio
4. The length and width of a rectangle are in a
What are the length and width?
. What are the measures of the angles?
ratio. The area of the rectangle is
5. Prove the converse of Theorem 7-1: For real numbers
,
,
, and
, with,
square inches.
and
.
Given:
,
,
, and
are real numbers, with
and
and
Prove:
6. Which of the following statements are true for all real numbers
?
a. If
408
then
.
,
,
, and
,
and
,
b. If
then
.
c. If
then
.
d. If
then
.
7. Solve each proportion for
.
a.
b.
c.
8. Shawna drove
miles and used
gallons of gas. At that rate, she would use
drive
miles. Write a proportion that could be used to find the value of
.
gallons of gas to
9. Solve the proportion you wrote in exercise 8. How much gas would Shawna expect to use to drive
miles?
ratio, with
10. Rashid, Leon, and Maria are partners in a company. They divide the profits in a
Rashid getting the largest share and Leon getting the smallest share. In
the company had a total profit
of
. How much profit did each person receive?
Answers
1.
a.
b.
c.
d.
2.
a.
b.
c.
d.
409
e.
f.
g.
3.
4.
,
,
inches and
inches
5.
Statement
A.
,
,
, and
, and
, with
and
Reason
A. Given
.
B. Multiplication Property of Equality
B.
C. Arithmetic
C.
D. Arithmetic
D.
E.
E.
, identity property of equality
6.
a. No
b. Yes
c. Yes
d. No
7.
a.
b.
or
c.
8.
9.
or equivalent
. At that rate she would use about
gallons of gas.
10. Rashid gets $800,000, Leon gets $400,000, and Maria gets $600,000.
410
Properties of Proportions
Learning Objectives
•
Prove theorems about proportions.
•
Recognize true proportions.
•
Use proportions theorems in problem solving.
Introduction
The Cross Multiplication Theorem is the basic, defining property of proportions. Whenever you are in doubt
about whether a proportion is true, you can always check it by cross multiplication. Additionally, there are
also a number of “sub-theorems” about proportions that are useful to apply for solving problems. In each
case the sub-theorem is easy to prove using cross multiplication.
Properties of Proportions
Technically speaking, the theorems in this lesson are not called sub-theorems. The formal term is corollary.
The word corollary is rather loosely defined in mathematics. Basically, a corollary is a theorem that follows
quickly, easily, and directly from another theorem—in this case from the Cross multiplication Theorem.
The corollaries in this section are not absolutely essential—you could always go back to using cross multiplication. But there may be times when the corollaries make things quicker or easier, so it’s good to have
them if and when they are needed.
Cross Multiplication Corollaries
Below are three corollaries that are immediate results of the Cross Multiplication Theorem and the fundamental laws of algebra.
Corollaries 1, 2, and 3 of The Cross Multiplication Theorem If
, and
, and
,
,
, then ....
1.
.
2.
.
3.
.
In words.
1. A true proportion is also true if you “swap” the “means.”
2. A true proportion is also true if you “swap” the “extremes.”
3. A true proportion is also true if you “flip” it upside down.
Example 1
411
Look at the diagram below.
Suppose we’re given that
We know
.
, since
Here are some other proportions that must also be true by corollaries 1-3.
Two Additional Corollaries to the Cross Multiplication Theorem
Here we have two more corollaries to the Cross Multiplication Theorem. The “if” part of these theorems is
the same as above. So the given in each proof remains the same too.
Corollary 4: If
,
,
, and
, and
, then
.
Proof.
Statement
Reason
1. Given
1.
2.
3.
412
,
,
, and
, and
2. Cross Multiplication Theorem
3. Distributive Property
4. Distributive Property
4.
5. Substitution
5.
6. Substitution
6.
7. Cross Multiplication Theorem
7.
This second theorem is nearly the same as the previous,
Corollary 5: If
,
,
, and
, and
, then
The proof of this corollary is in the Lesson Exercises.
Example 2
Suppose we’re given that
again, as in example 1.
Here are some other proportions that must also be true, and the theorems that guarantee them.
Corollary 4
Corollary 5
Corollary 4
Lesson Summary
Proportions were probably not new to you in this lesson; you may have studied them in previous courses.
What probably is new is the larger structure of theorems and corollaries that serve as tools for working with
proportions.
The most basic fact about proportions is the Cross Multiplication Theorem:
assuming
. The corollaries in this lesson are really just variations on the Cross Multiplication Theorem. They may be useful in problems, but we could always revert back to Cross Multiplication
if we had to.
Some people find proportions nice to work with, because there are so many different—and correct—ways
to write a given proportion, as you saw in the corollaries. It sometimes seems that you would really have to
413
work at it to write a proportion that is not equivalent to the proportion you are given!
Points to Consider
As we move ahead we will meet important concepts that require the use of ratios and proportions. Proportions
are mandatory for understanding the geometric meaning of similar. Later when we work with transformations
and scale factors, ratios will also be useful.
Finally, one proof of the Pythagorean Theorem relies on proportions.
Lesson Exercises
Given that
,
be true. Otherwise write “false.”
,
. For each of the following, write “true” if the proportion must
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. Prove: If
414
,
,
, then
.
12. Prove Corollary 5 to the Cross Multiplication Theorem.
Answers
1. False
2. True
3. False
4. True
5. True
6. False
7. True
8. True
9. True
10. False
11.
Statement
A.
Reason
A. Given
,
,
B. Cross Multiplication Theorem
B.
C. Distributive Property
C.
D. Substitution
D.
E. Distributive Property
E.
F. Cross Multiplication Theorem
F.
12.
Statement
A.
B.
C.
D.
E.
Reason
A. Given
,
,
B. Cross Multiplication Theorem
C. Distributive Property
D. Substitution
E. Distributive Property
F. Cross Multiplication Theorem
F.
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Similar Polygons
Learning Objectives
•
Recognize similar polygons.
•
Identify corresponding angles and sides of similar polygons from a statement of similarity.
•
Calculate and apply scale factors.
Introduction
Similar figures, rectangles, triangles, etc., have the same shape. Same shape, however, is not a precise
enough term for geometry. In this lesson, we’ll learn a precise definition for similar, and apply it to measures
of the sides and angles of similar polygons.
Similar Polygons
Look at the triangles below.
•
The triangles on the left are not similar because they are not the same shape.
•
The triangles in the middle are similar. They are all the same shape, no matter what their sizes.
•
The triangles on the right are similar. They are all the same shape, no matter how they are turned or
what their sizes.
Look at the quadrilaterals below.
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•
The quadrilaterals in the upper left are not similar because they are not the same shape.
•
The quadrilaterals in the upper right are similar. They are all the same shape, no matter what their sizes.
•
The quadrilaterals in the lower left are similar. They are all the same shape, no matter how they are
turned or what their sizes.
Now let’s get serious about what it means for figures to be similar. The rectangles below are all similar to
each other.
These rectangles are similar, but it’s not just because they’re rectangles. Being rectangles guarantees that
these figures all have congruent angles. But that’s not enough. You’ve seen lots of rectangles before, some
are long and narrow, others are more blocky and closer to square in shape.
The rectangles above are all the same shape. To convince yourself of this you could measure the length
and width of each rectangle. Each rectangle has a length that is exactly twice its width. So the ratio of lengthto-width is
for each rectangle. Now we can make a more formal statement of what similar means in
geometry.
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Two polygons are similar if and only if:
•
they have the same number of sides
•
for each angle in either polygon there is a corresponding angle in the other polygon that is congruent
•
the lengths of all corresponding sides in the polygons are proportional
Reminder: Just as we did with congruent figures, we name similar polygons according to corresponding
is used to represent “is similar to.” Some people call this “the congruent sign without
parts. The symbol
the equals part.”
Example 1
Suppose
Based on this statement, which angles are congruent and which sides are
proportional? Write true congruence statements and proportions.
,
, and
Remember that there are many equivalent ways to write a proportion. The answer above is not the only set
of true proportions you can create based on the given similarity statement. Can you think of others?
Example 2
Given:
What are the values of
,
, and
Set up a proportion to solve for
:
Now set up a proportion to solve for
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in the diagram below?
:
Finally, since
is an angle, we are looking for
Example 3
is a rectangle with length
and width
is a rectangle with length
and width
.
.
A. Are corresponding angles in the rectangles congruent?
Yes. Since both are rectangles, all the angles in both are congruent right angles.
B. Are the lengths of the sides of the rectangles proportional?
. The ratio of the widths is
No. The ratio of the lengths is
the lengths of the sides are not proportional.
. Therefore,
C. Are the rectangles similar?
No. Corresponding angles are congruent, but lengths of corresponding sides are not proportional.
Example 4
Prove that all squares are similar.
Our proof is a “paragraph” proof in bullet form, rather than a two-column proof:
Given two squares.
•
All the angles of both squares are right angles, so all angles of both squares are congruent—and this
includes corresponding angles.
•
Let the length of each side of one square be
, and the length of each side of the other square be
. Then the ratio of the length of any side of the first square to the length of any side of the second square
is
. So the lengths of the sides are proportional.
•
The squares satisfy the definition of similar polygons: congruent angles and proportional side lengths so they are similar
Scale Factors
If two polygons are similar, we know that the lengths of corresponding sides are proportional. If
is the
length of a side in one polygon, and
is the length of the corresponding side in the other polygon, then
the ratio
is called the scale factor relating the first polygon to the second. Another way to say this is:
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The length of every side of the first polygon is
polygon.
times the length of the corresponding side of the other
Example 5
Look at the diagram below, where
and
are similar rectangles.
A. What is the scale factor?
, then
.
Since
you know that
and
are corresponding sides. Since
is a rectangle,
The scale factor is the ratio of the lengths of any two corresponding sides.
So the scale factor (relating
to
) is
. We now know that the length of
is
times the length of the corresponding side in
.
each side of
Comment: We can turn this relationship around “backwards” and talk about the scale factor relating
to
to
. This scale factor is just
which is the reciprocal of the scale factor relating
B. What is the ratio of the perimeters of the rectangles?
is a
by
rectangle. Its perimeter is
.
is a
by
rectangle. Its perimeter is
.
The ratio of the perimeters of
to
is
.
Comment: You see from this example that the ratio of the perimeters of the rectangles is the same as the
scale factor. This relationship for the perimeters holds true in general for any similar polygons.
Ratio of Perimeters of Similar Polygons
Let’s prove the theorem that was suggested by example 5.
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Ratio of the Perimeters of Similar Polygons: If
and
are two similar polygons,
each with
sides and the scale factor of the polygons is
, then the ratio of the
perimeters of the polygons is .
•
Given:
and
are two similar polygons, each with
sides
The scale factor of the polygons is
•
Prove: The ratio of the perimeters of the polygons is
Statement
1.
and
are similar polygons, each with
2. The scale factor of the polygons is
3. Let
2. Given
be the lengths 3. Given (polygons have
each)
and
of corresponding sides of
sides
and
sides
4. Definition of scale factor
4.
5. Definition of perimeter
5. Perimeter of
6. Substitution
6.
7. Distributive Property
7.
8.
Reason
1. Given
8. Definition of perimeter
, the perimeter of
Comment: The ratio of the perimeters of any two similar polygons is the same as the scale factor. In fact,
the ratio of any two corresponding linear measures in similar figures is the same as the scale factor. This
applies to corresponding sides, perimeters, diagonals, medians, midsegments, altitudes, etc.
As we’ll see in an upcoming lesson, this is definitely not true for the areas of similar polygons. The ratio of
the areas of similar polygons (that are not congruent) is not the same as the scale factor.
Example 6
. The perimeter of
What is the perimeter of
is 150.
?
The scale factor relating
to
is
. According to the Ratio of the Perimeter's
Theorem, the perimeter of
is
of the perimeter of
. Thus, the perimeter of
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is
.
Lesson Summary
Similar has a very specific meaning in geometry. Polygons are similar if and only if the lengths of their sides
are proportional and corresponding angles are congruent. This is same shape translated into geometric
terms.
The ratio of the lengths of corresponding sides in similar polygons is called the scale factor. Lengths of other
corresponding linear measures, such as perimeter, diagonals, etc. have the same scale factor.
Points to Consider
Scale factors show the relationship between corresponding linear measures in similar polygons. The story
is not quite that simple for the relationship between the areas or volumes of similar polygons and polyhedra
(three-dimensional figures). We’ll study these relationships in future lessons.
Similar triangles are the basis for the study of trigonometry. The fact that the ratios of the lengths of corresponding sides in right triangles depends only on the measure of an angle, not on the size of the triangle,
makes trigonometric functions the property of an angle, as you will study in Chapter 8.
Lesson Exercises
True or false?
1. All equilateral triangles are similar.
2. All isosceles triangles are similar.
3. All rectangles are similar.
4. All rhombuses are similar.
5. All squares are similar.
6. All congruent polygons are similar.
7. All similar polygons are congruent.
Use the following diagram for exercises 8-11.
Given that rectangle
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rectangle
.
What is the value of each expression?
8.
9.
10.
11.
12. Given that
, what is the scale factor of the triangles?
Use the diagram below for exercises 13-16.
Given:
13. What is the perimeter of
14. What is the perimeter of
15. What is the ratio of the perimeter of
16. Prove:
17.
to the perimeter of
. [Write a flow proof.]
is the midpoint of
and
is the midpoint of
in
.
a. Name a pair of parallel segments.
b. Name two pairs of congruent angles.
c. Write a statement of similarity of two triangles.
d. If the perimeter of the larger triangle in c is
e. If the area of
is
, what is the perimeter of the smaller triangle?
, what is the area of quadrilateral
Answers
1. True
2. False
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3. False
4. False
5. True
6. True
7. False
8.
9.
10.
11.
12.
,
, or
,
,
13.
14.
15.
or equivalent
16.
, so the sides are all proportional.
(vertical angles)
,
(parallel lines, alternate interior angles are congruent)
(definition of similar polygons: angles are congruent, lengths of sides are proportional)
17.
a.
b.
c.
d.
e.
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or equivalent
Similarity by AA
Learning Objectives
•
Determine whether triangles are similar.
•
Understand AAA and AA rules for similar triangles.
•
Solve problems about similar triangles.
Introduction
You have an understanding of what similar polygons are and how to recognize them. Because triangles are
the most basic building block on which other polygons can be based, we now focus specifically on similar
triangles. We’ll find that there’s a surprisingly simple rule for triangles to be similar.
Angles in Similar Triangles
Tech Note - Geometry Software
Use your geometry software to experiment with triangles. Try this:
1. Set up two triangles,
and
.
2. Measure the angles of both triangles.
3. Move the vertices until the measures of the corresponding angles are the same in both triangles.
4. Compute the ratios of the lengths of the sides
.
Repeat steps 1-4 with different triangles. Observe what happens in step 4 each time. Record your observations.
What did you see during your experiment? You might have noticed this: When you adjust triangles to make
their angles congruent, you automatically make the sides proportional (the ratios in step 4 are the same).
Once we have triangles with congruent angles and sides with proportional lengths, we know that the triangles
are similar.
Conclusion: If the angles of a triangle are congruent to the corresponding angles of another triangle, then
the triangles are similar. This is a handy rule for similar triangles—a rule based on just the angles of the triangles. We call this the AAA rule.
Caution: The AAA rule is a rule for triangles only. We already know that other pairs of polygons can have
all corresponding angles congruent even though the polygons are not similar.
Example 1
The following is false statement: If the corresponding angles of two polygons are congruent, then the polygons
are similar.
What is a counterexample to the false statement above?
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Draw two polygons that are not similar, but which do have all corresponding angles congruent.
Rectangles such as the ones below make good examples.
Note: All rectangles have congruent (right) angles. However, we saw in an earlier lesson that rectangles
can have different shapes—long and narrow vs. stubby and square-ish. In formal terms, these rectangles
have congruent angles, but their side lengths are obviously not proportional. The rectangles are not similar.
Congruent angles are not enough to ensure similarity for rectangles.
The AA Rule for Similar Triangles
Some artists and designers apply the principle that “less is more.” This idea has a place in geometry as well.
Some geometry scholars feel that it is more satisfying to prove something with the least possible information.
Similar triangles are a good example of this principle.
The AAA rule was developed for similar triangles earlier. Let’s take another look at this rule, and see if we
can reduce it to “less” rather than “more.”
and
Suppose that triangles
and
have two pairs of congruent angles, say
.
But we know that if triangles have two pairs of congruent angles, then the third pair of angles are also congruent (by the Triangle Sum Theorem).
Summary: Less is more. The AAA rule for similar triangles reduces to the AA triangle similarity postulate.
The AA Triangle Similarity Postulate: If two pairs of corresponding angles in two triangles are congruent,
then the triangles are similar.
Example 2
Look at the diagram below.
A. Are the triangles similar? Explain your answer.
Yes. They both have congruent right angles, and they both have a
AA.
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angle. The triangles are similar by
B. Write a similarity statement for the triangles.
or equivalent
C. Name all pairs of congruent angles.
,
,
D. Write equations stating the proportional side lengths in the triangles.
or equivalent
Indirect Measurement
A traditional application of similar triangles is to measure lengths indirectly. The length to be measured would
be some feature that was not easily accessible to a person. This length might be:
•
the width of a river
•
the height of a tall object
•
the distance across a lake, canyon, etc.
To measure indirectly, a person would set up a pair of similar triangles. The triangles would have three
known side lengths and the unknown length. Once it is clear that the triangles are similar, the unknown
length can be calculated using proportions.
Example 3
Flo wants to measure the height of a windmill. She held a
foot vertical pipe with its base touching the level ground, and the pipe’s shadow was
feet long. At the same time, the shadow of the tower was
feet long. How tall is the tower?
Draw a diagram.
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Note: It is safe to assume that the sun’s rays hit the ground at the same angle. It is also proper to assume
that the tower is vertical (perpendicular to the ground).
The diagram shows two similar right triangles. They are similar because each has a right angle, and the
angle where the sun’s rays hit the ground is the same for both objects. We can write a proportion with only
one unknown,
, the height of the tower.
Thus, the tower is
feet tall.
Note: This is method considered indirect measurement because it would be difficult to directly measure the
height of tall tower. Imagine how difficult it would be to hold a tape measure up to a 51-foot-tall tower.
Lesson Summary
The most basic way—because it requires the least input of information—to assure that triangles are similar
is to show that they have two pairs of congruent angles. The AA postulate states this: If two triangles have
two pairs of congruent angles, then the triangles are similar.
Once triangles are known to be similar, we can write many true proportions involving the lengths of their
sides. These proportions were the basis for doing indirect measurement.
Points to Consider
Think about some right triangles for a minute. Suppose two right triangles both have an acute angle that
measures
. Then the ratio
is the same in both triangles. In fact, this ratio, called
“the tangent of
” is the same in any right triangle with a
angle. As mentioned earlier, this is the
reason for trigonometric functions of a given angle being constant, regardless of the specific triangle involved.
Lesson Exercises
Use the diagram below for exercises 1-5.
Given that
:
1. Name two similar triangles.
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2. Explain how you know that the triangles you named in exercise 1 are similar.
3. Write a true proportion.
4. Name two triangles that might not be similar.
5. If
,
6. Given that
Write an expression for
, and
,
, what is the length of
, and
in terms of
?
in the diagram below
.
7. Prove the following theorem:
If an acute angle of a right triangle is congruent to an acute angle of another right triangle, then the triangles
are congruent.
Write a flow proof.
Use the following diagram for exercises 8-12.
In a geometry reality competition, the teams must estimate the width of the river shown in the diagram.
Here’s what they did.
•
Anna, Bela, and Carlos stayed on the upper bank of the river.
•
Darryl and Eva paddled across to the lower bank of the river.
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•
Carlos placed a marker at
•
Darryl placed a marker directly across from Carlos at
•
Bela walked
at
.
feet back from the bank in a line with the markers at
•
Anna walked
feet on a path perpendicular to
•
Eva moved along the lower bank until she was lined up with
,
, and
.
.
and
and placed a marker at
and
are on land, so they can be measured easily.
and placed a marker
.
, and placed a marker at
was measured to be
.
feet.
8. Name two similar triangles.
9. Explain how you know that the triangles in exercise 8 are similar.
10. Write a proportion in which the only unknown measure is
.
11. How wide is the river?
12. Discuss whether or not the triangles used to answer exercises 8-11 are good models for a river and its
banks.
Answers
1.
,
or equivalent
2. The triangles have two pairs of congruent alternate interior angles and one pair of congruent vertical angles.
They are similar by AAA and AA.
3. Any proportion obtained from
4.
,
or
5.
6.
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,
,
for example
7. One acute angle in each triangle is congruent to an acute angle in the other triangle. Also, since they are
right triangles, both triangles have a right angle, and these right angles are congruent. The triangles are
congruent by AA.
8.
9.
and
are congruent right angles.
angles. The triangles are similar by AA.
:
, so
and
are congruent alternate interior
10.
11.
The river is approximately 133 feet wide.
12. This seems to be a good model. The banks are roughly straight enough to be lines. The banks appear
to be nearly parallel. If we can accept that parallel straight lines adequately represent the river banks, then
the model is a good one.
Similarity by SSS and SAS
Learning Objectives
•
Use SSS and SAS to determine whether triangles are similar.
•
Apply SSS and SAS to solve problems about similar triangles.
Introduction
You have been using the AA postulate to work with similar triangles. AA is easy to state and to apply. In
addition, there are other similarity postulates that should remind you of some of the congruence postulates.
These are the SSS and SAS similarity postulates. These postulates will give us more tools for recognizing
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similar triangles and solving problems involving them.
Exploring SSS and SAS for Similar Triangles
We’ll use geometry software and compass-and-straightedge constructions to explore relationships among
triangles based on proportional side lengths and congruent angles.
SSS for Similar Triangles
Tech Note - Geometry Software
Use your geometry software to explore triangles with proportional side lengths. Try this.
1.
Set up two triangles,
and
, with each side length of
length of the corresponding side of
being
times the
.
2. Measure the angles of both triangles.
3. Record the results in a chart like the one below.
Repeat steps 1-3 for each value of
in the chart. Keep
the same throughout the exploration.
Triangle Data
•
First, you know that all three side lengths in the two triangles are proportional. That’s what it means for
each side in
•
to be
times the length of the corresponding side in
.
. Each time you made a new
You probably notice what happens with the angle measures in
triangle
for the given value of
, the measures of
,
, and
were approximately
the same as the measures of
,
, and
. Like before when we experimented with the AA
and AAA relationships, there is something “automatic” that happens. If the lengths of the sides of the
triangles are proportional, that “automatically” makes the angles in the two triangles congruent too. Of
course, once we know that the angles are congruent, we also know that the triangles are similar by AAA
or AA.
Hands-On Activity
Materials: Ruler/straightedge, compass, protractor, graph or plain paper.
Directions: Work with a partner in this activity. Each partner will use tools to draw a triangle.
Each partner can work on a sheet of graph paper or on plain paper. Make drawings as accurate as possible.
Note that it doesn’t matter what unit of length you use.
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1. Partner 1: Draw a 6-8-10 triangle.
2. Partner 2: Draw a 9-12-15 triangle.
3. Partner 1: Measure the angles of your triangle.
4. Partner 2: Measure the angles of your triangle.
5. Partners 1 and 2: Compare your results.
What do you notice?
•
First, you know that all three side lengths in the two triangles are proportional.
•
You also probably noticed that the angles in the two triangles are congruent. You might want to repeat
the activity, drawing two triangles with proportional side lengths. You should find, again, that the angles
in the triangles are automatically congruent.
•
Once we know that the angles are congruent, then we know that the triangles are similar by AAA or AA.
SSS for Similar Triangles
Conclusion: If the lengths of the sides of two triangles are proportional, then the triangles are similar. This
is known as SSS for similar triangles.
SAS for Similar Triangles
SAS for Similar Triangles
If the lengths of two corresponding sides of two triangles are proportional and the included
angles are congruent, then the triangles are similar. This is known as SAS for similar triangles.
Example 1
Cheryl made the diagram below to investigate similar triangles more.
She drew
first, with
,
, and
.
Then Cheryl did the following:
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She drew
, and made
Then she carefully drew
.
, making
and
At this point, Cheryl had drawn two segments (
lengths of the corresponding sides of
to the included angle (
) in
=
and
.
) with lengths that are proportional to the
, and she had made the included angle,
, congruent
.
Then Cheryl measured angles. She found that:
•
•
What could Cheryl conclude? Here again we have automatic results. The other angles are automatically
congruent, and the triangles are similar by AAA or AA. Cheryl’s work supports the SAS for Similar Triangles
Postulate.
Similar Triangles Summary
We’ve explored similar triangles extensively in several lessons. Let’s summarize the conditions we’ve found
that guarantee that two triangles are similar.
Two triangles are similar if and only if:
•
the angles in the triangles are congruent.
•
the lengths of corresponding sides in the polygons are proportional.
AAA:: If the angles of a triangle are congruent to the corresponding angles of another triangle, then the triangles are similar.
AA:: It two pairs of corresponding angles in two triangles are congruent, then the triangles are similar.
SSS for Similar Triangles: If the lengths of the sides of two triangles are proportional, then the triangles
are similar.
SAS for Similar Triangles: If the lengths of two corresponding sides of two triangles are proportional and
the included angles are congruent, then the triangles are similar.
Points to Consider
Have you ever made a model rocket? Have you seen a scale drawing? Do you know people who use
blueprints? Do you enlarge pictures on your computer or shrink them? These are all examples of similar
two-dimensional or three-dimensional objects.
Lesson Exercises
Triangle 1 has sides with lengths
inches,
Triangle 2 has sides with lengths
feet,
inches, and
feet, and
inches.
feet.
1. Are Triangle 1 and Triangle 2 congruent? Explain your answer.
2. Are Triangle 1 and Triangle 2 similar? Explain your answer.
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3. What is the scale factor from Triangle 1 to Triangle 2?
4. Why do we not study an ASA similarity postulate?
Use the chart below for exercises 4-9.
Must
and
be similar?
5.
6.
7.
8.
9.
10.
11. Hands-On Activity
Materials: Ruler/straightedge, compass, protractor, graph or plain paper.
Directions: Work with a partner in this activity. Each partner will use tools to draw a triangle.
Each partner can work on a sheet of graph paper or on plain paper. Make drawings as accurate as possible.
Note that it doesn’t matter what unit of length you use.
with
Partner 1: Draw
Partner 2: Draw
A. Are sides
,
with
,
,
,
, and
=
, and
=
, and
.
proportional?
Partner 1: Measure the other angles of your triangle.
Partner 2: Measure the other angles of your triangle.
Partners 1 and 2: Compare your results.
B. Are the other angles of the two triangles (approximately) congruent?
C. Are the triangles similar? If they are, write a similarity statement and explain how you know that the triangles are similar.
.
Answers
1. No. One is much larger than the other.
2. Yes, SSS. The side lengths are proportional.
3.
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4. There is no need. With the
triangles are similar by AA.
and
parts of
we have triangles with two congruent angles. The
5. Yes
6. No
7. No
8. Yes
9. No
10. Yes
11.
A. Yes
B. Yes
C. Yes. All three pairs of angles are congruent, so the triangles are similar by AAA or AA.
Proportionality Relationships
Learning Objectives
•
Identify proportional segments when two sides of a triangle are cut by a segment parallel to the third
side.
•
Divide a segment into any given number of congruent parts.
Introduction
We’ll wind up our study of similar triangles in this section. We will also extend some basic facts about similar
triangles to dividing segments.
Dividing Sides of Triangles Proportionally
Think about a midsegment of a triangle. A midsegment is parallel to one side of a triangle, and that it divides
the other two sides into congruent halves (because the midsegment connects the midpoints of those two
sides). So the midsegment divides those two sides proportionally.
Example 1
Explain the meaning of "the midsegment divides the sides of a triangle proportionally."
Suppose each half of one side of a triangle is
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units long, and each half of the other side is
units long.
One side is divided in the ratio
to
and to each other.
, the other side in the ratio
Both of these ratios are equivalent
We see that a midsegment divides two sides of a triangle proportionally. But what about some other segment?
Tech Note - Geometry Software
Use your geometry software to explore triangles where a line parallel to one side intersects the other two
sides. Try this:
1. Set up
.
2. Draw a line that is parallel to
3. Label the intersection point on
and that intersects both of the other sides of
as
; label the intersection point on
.
as
.
Your triangle will look something like this.
parallel to
4. Measure lengths and calculate the following ratios.
______ and
_____
5. Compare your results with those of other students.
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Different students can start with different triangles. They can draw different lines parallel to
. But in
each case the two ratios,
and
, are approximately the same. This is another way to say that the
two sides of the triangle are divided proportionally. We can prove this result as a theorem.
Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects
the other two sides, then it divides those sides into proportional segments.
Proof.
•
Given:
with
•
Prove:
Statement
Reason
1. Given
1.
2.
2. Corresponding angles are congruent
3. AA Similarity Postulate
,
3.
4.
5.
,
4. Segment addition postulate
5. Corresponding side lengths in
similar triangles are proportional
6. Substitution
6.
7. Algebra
7.
8. Substitution
8.
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9. Addition property of equality
9.
Can you see why we wrote the proportion this way, rather than as
also a true proportion?
It’s because
, but there is no similar way to simplify
, which is
.
Note: The converse of this theorem is also true. If a line divides two sides of a triangle into proportional
segments, then the line is parallel to the third side of the triangle.
Example 2
In the diagram below,
What is an expression in terms of
.
for the length of
?
According to the Triangle Proportionality Theorem,
There are some very interesting corollaries to the Triangle Proportionality Theorem. One could be called
the Lined Notebook Paper Corollary!
Parallel Lines and Transversals
Example 3
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Look at the diagram below. We can make a corollary to the previous theorem.
are labels for lines
are lengths of segments
are parallel but not equally spaced
We’re given that lines
,
, and
are parallel. We can see that the parallel lines cut lines
and
(transversals). A corollary to the Triangle Proportionality Theorem states that the segment lengths on one
transversal are proportional to the segment lengths on the other transversal.
Conclusion:
and
Example 4
The corollary in example 3 can be broadened to any number of parallel lines that cut any number of
transversals. When this happens, all corresponding segments of the transversals are proportional!
The diagram below shows several parallel lines,
, and
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.
,
,
and
, that cut several transversals
,
lines are all parallel.
Now we have lots of proportional segments.
For example:
,
,
,
, and many more.
This corollary extends to more parallel lines cutting more transversals.
Lined Notebook Paper Corollary
Think about a sheet of lined notebook paper. A sheet has numerous equally spaced horizontal parallel
segments; these are the lines a person can write on. And there is a vertical segment running down the left
side of the sheet. This is the segment setting the margin, so you don’t write all the way to the edge of the
paper.
Now suppose we draw a slanted segment on the sheet of lined paper.
Because the vertical margin segment is divided into congruent parts, then the slanted segment is also divided
into congruent segments. This is the Lined Notebook Paper Corollary.
What we’ve done here is to divide the slanted segment into five congruent parts. By placing the slanted
segment differently we could divide it into any given number of congruent parts.
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History Note
In ancient times, mathematicians were interested in bisecting and trisecting angles and segments. Bisection
was no problem. They were able to use basic geometry to bisect angles and segments.
But what about trisection—dividing an angle or segment into exactly three congruent parts? This was a real
challenge! In fact, ancient Greek geometers proved that an angle cannot be trisected using only compass
and straightedge.
With the Lined Notebook Paper Corollary, though, we have an easy way to trisect a given segment.
Example 5
Trisect the segment below.
Draw equally spaced horizontal lines like lined notebook paper. Then place the segment onto the horizontal
lines so that its endpoints are on two horizontal lines that are three spaces apart.
•
slanted segment is same length as segment above picture
•
endpoints are on the horizontal segments shown
•
slanted segment is divided into three congruent parts
The horizontal lines now trisect the segment. We could use the same method to divide a segment into any
required number of congruent smaller segments.
Lesson Summary
In this lesson we began with the basic facts about similar triangles—the definition and the SSS and SAS
properties. Then we built on those to create numerous proportional relationships. First we examined proportional sides in triangles, then we extended that concept to dividing segments into proportional parts. We finalized those ideas with a notebook paper property that gave us a way to divide a segment into any given
number of equal parts.
Points to Consider
Earlier in this book you studied congruence transformations. These are transformations in which the image
is congruent to the original figure. You found that translations (slides), rotations (turns), and reflections (flips)
are all congruence transformations. In the next lesson we’ll study similarity transformations—transformations
in which the image is similar to the original figure. We’ll focus on dilations. These are figures that we zoom
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in on, or zoom out on. The idea is very similar to blowing up or shrinking a photo before printing it.
Lesson Exercises
Use the diagram below for exercises 1-5.
Given that
1. Name similar triangles.
Complete the proportion.
2.
3.
4.
5.
Lines
,
, and
are parallel.
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6. What is the value of
Lines
,
, and
?
are parallel, and
7. What is the value of
?
8. What is the value of
?
.
9. Explain how to divide a segment into seven congruent segments using the Lined Notebook Paper
Corollary.
Answers
1.
2.
3.
4.
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or equivalent
5.
6.
7.
8.
9. Place the original segment so that one endpoint is on the top horizontal line. Slant the segment so that
the other endpoint is on the seventh horizontal line below the top line. These eight horizontal lines divide
the original segment into seven congruent smaller segments.
Similarity Transformations
Learning Objectives
•
Draw a dilation of a given figure.
•
Plot the image of a point when given the center of dilation and scale factor.
•
Recognize the significance of the scale factor of a dilation.
Introduction
Earlier you studied one group of transformations that “preserve” length. This means that the image of a
segment is a congruent segment. These congruence transformations are translations, reflections, and
rotations.
In this lesson, you’ll study one more kind of transformation, the dilation. Dilations do not preserve length,
meaning the image of a segment can be a segment that is not congruent to the original. You’ll see that the
image of a figure in a dilation is a similar, not necessarily congruent, figure.
Dilations
A dilation is like a “blow-up” of a photo to change its size. A dilation may make a figure larger, or smaller,
but the same shape as the original. In other words, as you’ll see, a dilation gives us a figure similar to the
original.
A dilation is a transformation that has a center and a scale factor. The center is a point and the scale
factor governs how much the figure stretches or shrinks.
Think about watching a round balloon being inflated, and focusing on the point exactly in the middle of the
balloon. The balloon stretches outwards from this point uniformly. So for example, if a circle is drawn around
the point, this circle will grow as the balloon stretches away from the points.
Dilation with center at point
Given a point
collinear with
that is
and
and scale factor
units from point
and
units from
,
. The image of
for this dilation is the point
that is
, the center of dilation.
Example 1
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The center of dilation is
Point
is
to locate
,
units from
.
. To find the image of point
, the image of
, and
. Point
, we go
is three times as far (
units from
units) from
as
is (
along
units), and
are collinear.
Note: The scale factor is
to
, and the scale factor is
. The length from
to
is “stretched” three times as long as the length from
.
Example 2
The center of dilation is
, and the scale factor is
Point
units from
, as in example 1. To find the image of point
to locate
, the image of
, and
are collinear.
is
along
units), and
,
Note: The scale factor is
.
Example 3
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. The length from
. Point
to
.
is
times as far (
“shrinks” to
, we go
units) from
units from
as
times as long as the length from
is (
to
is a rectangle. What are the length, width, perimeter, and area of
Length
?
Width
Area
Perimeter
The center of a dilation is
, and the scale factor is
length × width
. What are the length, width, perimeter, and area
of
Point
is the same as point
.
is
, and
is
.
In
Length
Width
Area
Perimeter
Note: The perimeter of
times the area of
is
length × width
times the perimeter of
, but the area of
is
.
As the following diagram shows, four rectangles congruent to
fit exactly into
.
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Coordinate Notation for Dilations
We can work with dilations on a coordinate grid. To simplify our work, we’ll study dilations that have their
center of dilation at the origin.
Triangle
Triangle
in the diagram below is dilated with scale factor
is the image of
Notice that each side of
.
is
times as long as the corresponding side of
that
,
, and the origin are collinear. Thus is also true of
and the origin.
This leads to the following generalization
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.
,
. Notice also
, and the origin, and of
,
,
Generalization: Points
,
(the image of
), and the origin are collinear for any point
You can prove the generalization in the Lesson Exercises.
in a dilation.
How do we know that a dilation is a similarity transformation? We would have to establish that lengths of
segments are proportional and that angles are congruent. Let’s attack these requirements through the distance
formula and slopes.
Let
,
and scale factor
, and
be points in a coordinate grid. Let a dilation have center at the origin
.
Part 1: Proportional Side Lengths
Let’s look at the lengths of two segments,
, and
.
According to the distance formula,
and
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What does this say about a segment and its image in a dilation? It says that the image of a segment is another
segment
times the length of the original segment. If a polygon had several sides, each side of the image
polygon would be
times the length of its corresponding side in the original polygon.
Conclusion: If a polygon is dilated, the corresponding sides of the image polygon and the original polygon
are proportional. So half the battle is over.
Part 2: Congruent Angles
Let’s look at the slopes of the sides of two angles,
and
.
slope of
slope of
slope of
slope of
Since
and
have the same slope, they are parallel. The same is true for
and
know that if the sides of two angles are parallel, then the angles are congruent. This gives us:
. We
Conclusion: If a polygon is dilated, the corresponding angles of the image polygon and the original polygon
are congruent. So the battle is now over.
Final Conclusion: If a polygon is dilated, the original polygon and the image polygon are similar, because
they have proportional side lengths and congruent angles. A dilation is a similarity transformation.
Lesson Summary
Dilations round out our study of geometric transformations. Unlike translations, rotations, and reflections,
dilations are not congruence transformations. They are similarity transformations. If a dilation is applied to
a polygon, the image is a similar polygon.
Points to Consider
We limited our study of dilations to those that have positive scale factors. To explore further, you might experiment with negative scale factors.
Tech Note - Geometry Software
Use your geometry software to explore dilations with negative scale factors.
Exploration 1
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•
Plot two points.
•
Select one of the points as the center of dilation.
•
Use 2 for the scale factor.
•
Find the image of the other point.
Repeat, but use a different value for the scale factor.
What seems to be true about the two images?
Exploration 2
•
Draw a triangle.
•
Select a point as the center of dilation. Use one vertex of the triangle, or draw another point for the
center.
•
Use 2 for the scale factor.
•
Find the image of the triangle.
•
Repeat, but use a different value for the scale factor.
What seems to be true about the two images?
You can experiment further with different figures, centers, and scale factors.
Can you reach any conclusions about images when the scale factor is negative?
You may have noticed that if point
is dilated, the center is
, and the scale factor is
,
, then
the image of
is on the same side of
as
is. If the scale factor is
then the image of
is on
. You may have also also noticed that a dilation with a negative scale factor is
the opposite side of
equivalent to a dilation with a positive scale factor followed by a “reflection in a point,” where the point is the
center of dilation.
This lesson brings our study of similar figures almost to a close. We’ll revisit similar figures once more in
Chapter 10, where we analyze the perimeter and area of similar polygons. Some writers have used similarity
concepts to explain why living things are the “right size” and why, for example, there are no
-foot-tall
human giants!
Lesson Exercises
Use the diagram below for exercises 1 - 10.
and
A dilation has the indicated center and scale factor. Complete the table.
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Center
Scale Factor
Given Point
Image of Answer
Given Point
1.
?
2.
?
3.
?
4.
?
5.
?
6.
?
7.
?
8.
?
or
9.
?
Midpoint of
or
10.
?
Midpoint of
11. Copy the square shown below. Draw the image of the square for a dilation with center at the intersection
of
and
scale factor
.
12. A given dilation is a congruence transformation. What is the scale factor of the dilation?
13. Imagine a dilation with a scale factor of
14. Let
collinear.
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and
. Describe the image of a given point for this dilation.
be two points in a coordinate grid,
. Prove that
and
are
15. A dilation has its center at the origin and a scale factor of
, and
is the image of
. Let
, what are the coordinates of
be the point . If
is the image of
?
Answers
11.
Small square centered in big square, each side of big square
times side of small square
12.
13. The image of any point is the point that is the center of dilation.
14.
Let
be the origin
.
slope of
slope of
slope of
,
Since the segments have common endpoints and the same slope, they are collinear.
15.
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8. Right Triangle Trigonometry
The Pythagorean Theorem
Learning Objectives
•
Identify and employ the Pythagorean Theorem when working with right triangles.
•
Identify common Pythagorean triples.
•
Use the Pythagorean Theorem to find the area of isosceles triangles.
•
Use the Pythagorean Theorem to derive the distance formula on a coordinate grid.
Introduction
The triangle below is a right triangle.
The sides labeled
and
are called the legs of the triangle and they meet at the right angle. The third
side, labeled is called the hypotenuse. The hypotenuse is opposite the right angle. The hypotenuse of
a right triangle is also the longest side.
The Pythagorean Theorem states that the length of the hypotenuse squared will equal the sum of the squares
of the lengths of the two legs. In the triangle above, the sum of the squares of the legs is
square of the hypotenuse is
.
The Pythagorean Theorem: Given a right triangle with legs whose length is
and a hypotenuse of length
and the
,
and
.
Be careful when using this theorem—you must make sure that the legs are labeled
and
and the hypotenuse is labeled to use this equation. A more accurate way to write the Pythagorean Theorem is:
Example 1
Use the side lengths of the following triangle to test the Pythagorean Theorem.
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The legs of the triangle above are inches and inches. The hypotenuse is inches. So,
,
, and
. We can substitute these values into the formula for the Pythagorean Theorem to verify that
the relationship works:
Since both sides of the equation equal
worked on this right triangle.
, the equation is true. Therefore, the Pythagorean Theorem
Proof of the Pythagorean Theorem
There are many ways to prove the Pythagorean Theorem. One of the most straightforward ways is to use
similar triangles. Start with a right triangle and construct an altitude from the right angle to the opposite sides.
In the figure below, we can see the following relationships:
Proof.
•
•
Given:
as shown in the figure below
Prove:
First we start with a triangle similarity statement:
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These are all true by the
triangle similarity postulate.
Now, using similar triangles, we can set up the following proportion:
and
Putting these equations together by using substitution,
factoring the right hand side,
but notice
, so this becomes
.
We have finished proving the Pythagorean Theorem. There are hundreds of other ways to prove the
Pythagorean Theorem and one of those alternative proofs is in the exercises for this section.
Making Use of the Pythagorean Theorem
As you know from algebra, if you have one unknown variable in an equation, you can solve to find its value.
Therefore, if you know the lengths of two out of three sides in a right triangle, you can use the Pythagorean
Theorem to find the length of the missing side, whether it is a leg or a hypotenuse. Be careful to use inverse
operations properly and avoid careless mistakes.
Example 2
What is the length of
in the triangle below?
Use the Pythagorean Theorem to find the length of the missing leg, . Set up the equation
, letting
and
. Be sure to simplify the exponents and roots carefully, remember to use inverse
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operations to solve the equation, and always keep both sides of the equation “balanced”.
In algebra you learned that
because, for example,
. However, in this
case (and in much of geometry), we are only interested in the positive solution to
metric lengths are positive. So, in example 2, we can disregard the solution
is
inches.
because geo, and our final answer
Example 3
Find the length of the missing side in the triangle below.
Use the Pythagorean Theorem to set up an equation and solve for the missing side. Let
.
and
So, the length of the missing side is 13 centimeters.
Using Pythagorean Triples
In example 1, the sides of the triangle were , , and . This combination of numbers is referred to as a
Pythagorean triple. A Pythagorean triple is three numbers that make the Pythagorean Theorem true and
they are integers (whole numbers with no decimal or fraction part). Throughout this chapter, you will use
other Pythagorean triples as well. For instance, the triangle in example 2 is proportional to the same ratio
of
. If you divide the lengths of the triangle in example 2
by two, you find the same
. Whenever you find a Pythagorean triple, you can apply those ratios with greater
proportion—
factors as well. Finally, take note of the side lengths of the triangle in example 3—
. This, too,
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is a Pythagorean triple. You can extrapolate that this ratio, multiplied by greater factors, will also yield
numbers that satisfy the Pythagorean Theorem.
There are infinitely many Pythagorean triples, but a few of the most common ones and their multiples are:
Triple
Area of an Isosceles Triangle
There are many different applications of the Pythagorean Theorem. One way to use The Pythagorean
Theorem is to identify the heights in isosceles triangles so you can calculate the area. The area of a triangle
is half of the product of its base and its height (also called altitude). This formula is shown below.
If you are given the base and the sides of an isosceles triangle, you can use the Pythagorean Theorem to
calculate the height. Recall that the height (altitude) of a triangle is the length of a segment from one angle
in the triangle perpendicular to the opposite side. In this case we focus on the altitude of isosceles triangles
going from the vertex angle to the base.
Example 4
What is the height of the triangle below?
To find the area of this isosceles triangle, you will need to know the height in addition to the base. Draw in
the height by connecting the vertex of the triangle with the base at a right angle.
Since the triangle is isosceles, the altitude will bisect the base. That means that it will divide it into two
congruent, or equal parts. So, you can identify the length of one half of the base as centimeters.
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If you look at the smaller triangle now inscribed in the original shape, you’ll notice that it is a right triangle
with one leg and hypotenuse
So, this is a
triangle. If the leg is cm and the hypotenuse is
cm, the missing leg must be cm. So, the height of the isosceles triangle is cm.
Use this information along with the original measurement of the base to find the area of the entire isosceles
triangle.
2
The area of the entire isosceles triangle is 12cm .
The Distance Formula
You have already learned that you can use the Pythagorean Theorem to understand different types of right
triangles, find missing lengths, and identify Pythagorean triples. You can also apply the Pythagorean Theorem
to a coordinate grid and learn how to use it to find distances between points.
Example 5
Look at the points on the grid below.
Find the length of the segment connecting
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and
.
The question asks you to identify the length of the segment. Because the segment is not parallel to either
axis, it is difficult to measure given the coordinate grid. However, it is possible to think of this segment as
the hypotenuse of a right triangle. Draw a vertical line at
and a horizontal line at
point of intersection. This point represents the third vertex in the right triangle.
and find the
You can easily count the lengths of the legs of this triangle on the grid. The vertical leg extends from
to
, so it is
units long. The horizontal leg extends from
to
, so it is
units long. Use the Pythagorean Theorem with these values for the lengths of each leg
to find the length of the hypotenuse.
The segment connecting
and
is
units long.
Mathematicians have simplified this process and created a formula that uses these steps to find the distance
between any two points in the coordinate plane. If you use the distance formula, you don’t have to draw the
extra lines.
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Distance Formula: Give points
and
, the length of the segment con-
necting those two points is
Example 6
Use the distance formula
and (5,2) on a coordinate grid.
to find the distance between the points (1,5)
You already know from example 1 that the distance will be units, but you can practice using the distance
formula to make sure it works. In this formula, substitute for
, for
, for
, and for
because
and
are the two points in question.
Now you see that no matter which method you use to solve this problem, the distance between
on a coordinate grid is
and
units.
Lesson Summary
In this lesson, we explored how to work with different radical expressions both in theory and in practical situations. Specifically, we have learned:
•
How to identify and employ the Pythagorean Theorem when working with right triangles.
•
How to identify common Pythagorean triples.
•
How to use the Pythagorean Theorem to find the area of isosceles triangles.
•
How to use the Pythagorean Theorem to derive the distance formula on a coordinate grid.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to apply the Pythagorean Theorem to mathematical situations.
Points to Consider
Now that you have learned the Pythagorean Theorem, there are countless ways to apply it. Could you use
the Pythagorean Theorem to prove that a triangle contained a right angle if you did not have an accurate
diagram?
Lesson Exercises
1. What is the distance between
2. Do the numbers
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,
, and
and
?
make a Pythagorean triple?
3. What is the length of
4. Do the numbers
,
in the triangle below?
, and
make a Pythagorean triple?
5. What is the distance between
6. What is the length of
and
in the triangle below?
7. What is the distance between
8. What is the area of
?
and
?
below?
9. What is the area of the triangle below?
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10. What is the area of the triangle below?
11. An alternative proof of the Pythagorean Theorem uses the area of a square. The diagram below shows
a square with side lengths
, and an inner square with side lengths
. Use the diagram below to
prove
Hint: Find the area of the inner square in two ways: once directly, and once by finding the area of the larger
square and subtracting the area of each triangle.
Answers
1.
[Diff: 1]
2. yes [Diff: 1]
3.
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inches [Diff: 2]
4. no [Diff: 1]
5.
[Diff: 2]
6.
inches [Diff: 2]
7.
[Diff: 2]
8.
square millimeters [Diff: 3]
9.
square feet [Diff: 3]
10.
square yards [Diff: 3]
11. Proof. The plan is, we will find the area of the green square in two ways. Since those two areas must
be equal, we can set those areas equal to each other.
For the inner square (in green), we can directly compute the area:
Now, the area of the large, outer square is
.
. Don’t forget to multiply this binomial carefully!
The area of each small right triangle (in yellow) is
.
Since there are four of those right triangles, we have the combined area
Finally, subtract the area of the four yellow triangles from the area of the larger square, and we are left with
Putting together the two different ways for finding the area of the inner square, we have
Converse of the Pythagorean Theorem
Learning Objectives
•
Understand the converse of the Pythagorean Theorem.
469
•
Identify acute triangles from side measures.
•
Identify obtuse triangles from side measures.
•
Classify triangles in a number of different ways.
Converse of the Pythagorean Theorem
In the last lesson, you learned about the Pythagorean Theorem and how it can be used. As you recall, it
states that the sum of the squares of the legs of any right triangle will equal the square of the hypotenuse.
If the lengths of the legs are labeled and , and the hypotenuse is , then we get the familiar equation:
The Converse of the Pythagorean Theorem is also true. That is, if the lengths of three sides of a triangle
make the equation
true, then they represent the sides of a right triangle.
With this converse, you can use the Pythagorean Theorem to prove that a triangle is a right triangle, even
if you do not know any of the triangle’s angle measurements.
Example 1
Does the triangle below contain a right angle?
This triangle does not have any right angle marks or measured angles, so you cannot assume you know
whether the triangle is acute, right, or obtuse just by looking at it. Take a moment to analyze the side lengths
and see how they are related. Two of the sides
and
are relatively close in length. The third side
is about half the length of the two longer sides.
To see if the triangle might be right, try substituting the side lengths into the Pythagorean Theorem to see
if they makes the equation true. The hypotenuse is always the longest side, so
should be substituted
for . The other two values can represent and and the order is not important.
Since both sides of the equation are equal, these values satisfy the Pythagorean Theorem. Therefore, the
triangle described in the problem is a right triangle.
In summary, example 1 shows how you can use the converse of the Pythagorean Theorem. The Pythagorean
Theorem states that in a right triangle with legs
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and
, and hypotenuse
,
. The converse
of the Pythagorean Theorem states that if
, then the triangle is a right triangle.
Identifying Acute Triangles
Using the converse of the Pythagorean Theorem, you can identify whether triangles contain a right angle
or not. However, if a triangle does not contain a right angle, you can still learn more about the triangle itself
by using the formula from Pythagorean Theorem. If the sum of the squares of the two shorter sides of a triangle is greater than the square of the longest side, the triangle is acute (all angles are less than
). In
symbols, if
then the triangle is acute.
Identifying the "shorter" and "longest" sides may seem ambiguous if sides have the same length, but in this
case any ordering of equal length sides leads to the same result. For example, an equilateral triangle always
satisfies
and so is acute.
Example 2
Is the triangle below acute or right?
The two shorter sides of the triangle are and
sum of the squares of the two shorter legs.
The sum of the squares of the two shorter legs is
. The longest side of the triangle is
. First find the
Compare this to the square of the longest side,
The square of the longest side is
Since
triangle. Compare the two values to identify which is greater.
, this triangle is not a right
The sum of the square of the shorter sides is greater than the square of the longest side. Therefore, this is
an acute triangle.
Identifying Obtuse Triangles
As you have probably figured out, you can prove a triangle is obtuse (has one angle larger than
) by
using a similar method. Find the sum of the squares of the two shorter sides in a triangle. If this value is less
than the square of the longest side, the triangle is obtuse. In symbols, if
, then the triangle is
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obtuse. You can solve this problem in a manner almost identical to example 2 above.
Example 3
Is the triangle below acute or obtuse?
The two shorter sides of the triangle are
of the squares of the two shorter legs.
and
The longest side of the triangle is
The sum of the squares of the two shorter legs is
The square of the longest side is 100. Since
the two values to identify which is greater.
First find the sum
Compare this to the square of the longest side,
, this triangle is not a right triangle. Compare
Since the sum of the square of the shorter sides is less than the square of the longest side, this is an obtuse
triangle.
Triangle Classification
Now that you know the ideas presented in this lesson, you can classify any triangle as right, acute, or obtuse
given the length of the three sides. Begin by ordering the sides of the triangle from smallest to largest, and
substitute the three side lengths into the equation given by the Pythagorean Theorem using
Be sure to use the longest side for the hypotenuse.
•
•
•
If
, the figure is a right triangle.
If
, the figure is an acute triangle.
If
, the figure is an obtuse triangle.
Example 4
Classify the triangle below as right, acute, or obtuse.
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.
The two shorter sides of the triangle are and
sum of the squares of the two shorter legs.
. The longest side of the triangle is
The sum of the squares of the two shorter legs is
The square of the longest side is
. First find the
Compare this to the square of the longest side,
Therefore, the two values are not equal,
angle is not a right triangle. Compare the two values,
and
and this tri-
to identify which is greater.
Since the sum of the square of the shorter sides is greater than the square of the longest side, this is an
acute triangle.
Example 5
Classify the triangle below as right, acute, or obtuse.
The two shorter sides of the triangle are
sum of the squares of the two shorter legs.
The sum of the squares of the two legs is
and
. The longest side of the triangle is
. First find the
. Compare this to the square of the longest side,
.
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The square of the longest side is
right triangle.
Since these two values are equal,
, and this is a
Lesson Summary
In this lesson, we explored how to work with different radical expressions both in theory and in practical situations. Specifically, we have learned:
•
How to use the converse of the Pythagorean Theorem to prove a triangle is right.
•
How to identify acute triangles from side measures.
•
How to identify obtuse triangles from side measures.
•
How to classify triangles in a number of different ways.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to apply the Pythagorean Theorem and its converse to mathematical situations.
Points to Consider
Use the Pythagorean Theorem to explore relationships in common right triangles. Do you find that the sides
are proportionate?
Lesson Exercises
Solve each problem.
For exercises 1-8, classify the following triangle as acute, obtuse, or right based on the given side lengths.
Note, the figure is not to scale.
1.
2.
3.
4.
5.
6.
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7.
8.
9. In the triangle below, which sides should you use for the legs (usually called sides
hypotenuse (usually called side ), in the Pythagorean theorem? How do you know?
, and
) and the
10.
a.
b.
Answers
1. Right [Diff: 1]
2. Acute [Diff: 1]
3. Obtuse [Diff: 1]
4. Acute [Diff: 2]
5. Right [Diff: 2]
6. Acute [Diff: 2]
7. Obtuse [Diff: 2]
8. Obtuse [Diff: 3]
9. The side with length
does not matter [Diff: 3].
10.
,
should be the hypotenuse since it is the longest side. The order of the legs
[Diff: 3]
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Using Similar Right Triangles
Learning Objectives
•
Identify similar triangles inscribed in a larger triangle.
•
Evaluate the geometric mean of various objects.
•
Identify the length of an altitude using the geometric mean of a separated hypotenuse.
•
Identify the length of a leg using the geometric mean of a separated hypotenuse.
Introduction
In this lesson, you will study figures inscribed, or drawn within, existing triangles. One of the most important
types of lines drawn within a right triangle is called an altitude. Recall that the altitude of a triangle is the
perpendicular distance from one vertex to the opposite side. By definition each leg of a right triangle is an
altitude. We can find one more altitude in a right triangle by adding an auxiliary line segment that connects
the vertex of the right angle with the hypotenuse, forming a new right angle.
You may recall this is the figure that we used to prove the Pythagorean Theorem. In right triangle
above, the segment
to the hypotenuse
is an altitude. It begins at angle
, which is a right angle, and it is perpendicular
. In the resulting figure, we have three right triangles, and all of them are similar.
Inscribed Similar Triangles
You may recall that if two objects are similar, corresponding angles are congruent and their sides are proportional in length. In other words, similar figures are the same shape, but different sizes. To prove that two
triangles are similar, it is sufficient to prove that all angle measures are congruent (note, this is NOT true for
other polygons. For example, both squares and “long” rectangles have all
angles, but they are not
similar). Use logic, and the information presented above to complete Example 1.
Example 1
Justify the statement that
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.
In the figure above, the big triangle
and
,
. So, if
, and
is a right triangle with right angle
,
, and
and
are similar, they will all have angles of
.
First look at
.
always equals
, the missing angle,
, and
. Since the sum of the three angles in a triangle
, must measure
Lining up the congruent angles, we can write
Now look at
.
has a measure of
angles in a triangle always equals
, since
.
.
, and
. Since the sum of the three
, the missing angle,
, must measure
, since
Now, since the triangles have congruent corresponding angles,
and
are similar.
Thus,
. Their angles are congruent and their sides are proportional.
Note that you must be very careful to match up corresponding angles when writing triangle similarity statements. Here we should write
triangles are overlapping.
. This example is challenging because the
Geometric Means
When someone asks you to find the average of two numbers, you probably think of the arithmetic mean
(average). Chances are good you’ve worked with arithmetic means for many years, but the concept of a
geometric mean may be new. An arithmetic mean is found by dividing the sum of a set of numbers by the
number of items in the set. Arithmetic means are used to calculate overall grades and many other applications.
The big idea behind the arithmetic mean is to find a “measure of center” for a group of numbers.
A geometric mean applies the same principles, but relates specifically to size, length, or measure. For example, you may have two line segments as shown below. Instead of adding and dividing, you find a geometric
mean by multiplying the two numbers, then finding the square root of the product.
To find the geometric mean of these two segments, multiply the lengths and find the square root of the
product.
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So, the geometric mean of the two segments would be a line segment that is
concepts and strategies to complete example 2.
cm in length. Use these
Example 2
In
below, what is the geometric mean of
and
?
When finding a geometric mean, you first find the product of the items involved. In this case, segment
is
inches and segment
is inches. Then find the square root of this product.
So, the geometric mean of
and
in
is
inches.
Altitude as Geometric Mean
In a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean of
the lengths of the two segments of the hypotenuse. In the diagram below we can use
to create the proportion
. Solving for
,
.
You can use this relationship to find the length of the altitude if you know the length of the two segments of
the divided hypotenuse.
Example 3
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What is the length of the altitude
in the triangle below?
To find the altitude of this triangle, find the geometric mean of the two segments of the hypotenuse. In this
case, you need to find the geometric mean of and . To find the geometric mean, find the product of the
two numbers and then take its square root.
So
feet, or approximately
feet.
Example 4
What is the length of the altitude in the triangle below?
The altitude of this triangle is
. Remember the altitude does not always go “down”! To find
, find
the geometric mean of the two segments of the hypotenuse. Make sure that you fill in missing information
in the diagram. You know that the whole hypotenuse,
need to know
subtract.
So
is
, the length of the longer subsection of
inches long and
inches, but you
, to find the geometric mean. To do this,
inches. Write this measurement on the diagram to keep track of your work.
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Now find the geometric mean of
and
The altitude of the triangle will measure
to identify the length of the altitude.
inches.
Leg as Geometric Mean
Just as we used similar triangles to create a proportion using the altitude, the lengths of the legs in right triangles can also be found with a geometric mean with respect to the hypotenuse. The length of one leg in
a right triangle is the geometric mean of the adjacent segment and the entire hypotenuse. The diagram below
shows the relationships.
You can use this relationship to find the length of the leg if you know the length of the two segments of the
divided hypotenuse.
Example 5
What is the length of
480
in the triangle below?
To find
, the leg of the large right triangle, find the geometric mean of the adjacent segments of the hypotenuse and the entire hypotenuse. In this case, you need to find the geometric mean of and
. To
find the geometric mean, find the product of the two numbers and then take the square root of that product.
So,
millimeters or approximately
millimeters.
Example 6
If
, what is the value
in the triangle below?
To find
in this triangle, find the geometric mean of the adjacent segment of the hypotenuse and the
entire hypotenuse. Make sure that you fill in missing information in the diagram. You know that the two
shorter sections of the hypotenuse are
inches and inches, but you need to know the length of the
entire hypotenuse to find the geometric mean. To do this, add.
So,
inches. Write this measurement on the diagram to keep track of your work.
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Now find the geometric mean of
So,
and
to identify the length of the altitude.
inches.
Lesson Summary
In this lesson, we explored how to work with different radicals both in theory and in practical situations.
Specifically, we have learned:
•
How to identify similar triangles inscribed in a larger triangle.
•
How to evaluate the geometric mean of various objects.
•
How to identify the length of an altitude using the geometric mean of a separated hypotenuse.
•
How to identify the length of a leg using the geometric mean of a separated hypotenuse.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Points to Consider
How can you use the Pythagorean Theorem to identify other relationships between sides in triangles?
Lesson Exercises
1. Which triangles in the diagram below are similar?
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2. What is the geometric mean of two line segments that are
and
inches,respectively?
3. What is the geometric mean of two line segments that are
cm each?
4. Which triangles in the diagram below are similar?
5. What is the length of the altitude,
6. What is the length of
, in the triangle below?
in the triangle below?
7. What is the geometric mean of two line segments that are
yards and
yards, respectively?
8. What is the length of the altitude in the triangle below?
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Use the following diagram from exercises 9-11:
9.
= ____
10.
= ____
11.
= ____ (for an extra challenge, find
in two different ways)
12. What is the length of the altitude in the triangle below?
Answers
1. Triangles
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2.
inches [Diff: 2]
3.
cm [Diff: 1]
and
are all similar [Diff: 1].
4. Triangles
5.
and
inches [Diff: 2]
6.
mm, or approximately
7.
yards, or approximately
8.
feet, or approximately
9.
are all similar. [Diff: 2]
mm [Diff: 2]
yards [Diff: 2]
feet [Diff: 3]
inches, or approximately
inches [Diff: 3]
10.
inches or approximately
inches [Diff: 3]
11.
inches or approximately
inches. One way to find
is with the geometric mean:
inches. Alternatively, using the answer from 9 and one of the smaller right triangles,
inches [Diff: 3].
Special Right Triangles
Learning Objectives
•
Identify and use the ratios involved with right isosceles triangles.
•
Identify and use the ratios involved with
•
Identify and use ratios involved with equilateral triangles.
•
Employ right triangle ratios when solving real-world problems.
triangles.
Introduction
What happens when you cut an equilateral triangle in half using an altitude? You get two right triangles.
What about a square? If you draw a diagonal across a square you also get two right triangles. These two
right triangles are special special right triangles called the
and the
right triangles. They have unique properties and if you understand the relationships between the sides and
angles in these triangles, you will do well in geometry, trigonometry, and beyond.
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Right Isosceles Triangles
The first type of right triangle to examine is isosceles. As you know, isosceles triangles have two sides that
are the same length. Additionally, the base angles of an isosceles triangle are congruent as well. An
isosceles right triangle will always have base angles that each measure
and a vertex angle that measures
.
Don’t forget that the base angles are the angles across from the congruent sides. They don’t have to be on
the bottom of the figure.
Because the angles of all
triangles will, by definition, remain the same, all
triangles are similar, so their sides will always be proportional. To find the relationship between the sides,
use the Pythagorean Theorem.
Example 1
The isosceles right triangle below has legs measuring
centimeter.
Use the Pythagorean Theorem to find the length of the hypotenuse.
Since the legs are
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centimeter each, substitute
for both
and
, and solve for
:
In this example
cm.
What if each leg in the example above was
If each leg is
cm? Then we would have
cm, then the hypotenuse is
When the length of each leg was
cm.
, the hypotenuse was
. When the length of each leg was
, the
hypotenuse was
. Is this a coincidence? No. Recall that the legs of all
triangles are
proportional. The hypotenuse of an isosceles right triangle will always equal the product of the length of one
leg and
. Use this information to solve the problem in example 2.
Example 2
What is the length of the hypotenuse in the triangle below?
Since the length of the hypotenuse is the product of one leg and
One leg is
inches, so the hypotenuse will be
inches, or about
, you can easily calculate this length.
inches.
Equilateral Triangles
Remember that an equilateral triangle has sides that all have the same length. Equilateral triangles are also
equiangular—all angles have the same measure. In an equilateral triangle, all angles measure exactly
.
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Notice what happens when you divide an equilateral triangle in half.
When an equilateral triangle is divided into two equal parts using an altitude, each resulting right triangle is
a
triangle. The hypotenuse of the resulting triangle was the side of the original, and the
shorter leg is half of an original side. This is why the hypotenuse is always twice the length of the shorter
leg in a
triangle. You can use this information to solve problems about equilateral triangles.
30º-60º-90º Triangles
Another important type of right triangle has angles measuring
,
, and
. Just as you found a
constant ratio between the sides of an isosceles right triangle, you can find constant ratios here as well. Use
the Pythagorean Theorem to discover these important relationships.
Example 3
Find the length of the missing leg in the following triangle. Use the Pythagorean Theorem to find your answer.
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Just like you did for
triangles, use the Pythagorean theorem to find the missing side. In
this diagram, you are given two measurements: the hypotenuse
cm. Find the length of the missing leg
is
cm and the shorter leg
is
.
You can leave the answer in radical form as shown, or use your calculator to find the approximate value of
cm.
On your own, try this again using a hypotenuse of feet. Recall that since the
triangle
comes from an equilateral triangle, you know that the length of the shorter leg is half the length of the hypotenuse.
Now you should be able to identify the constant ratios in
triangles. The hypotenuse will
always be twice the length of the shorter leg, and the longer leg is always the product of the length of the
shorter leg and
.
. In ratio form, the sides, in order from shortest to longest are in the ratio
Example 4
What is the length of the missing leg in the triangle below?
Since the length of the longer leg is the product of the shorter leg and
length. The short leg is
inches, so the longer leg will be
, you can easily calculate this
inches, or about
inches.
Example 5
What is
below?
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To find the length of segment
, identify its relationship to the rest of the triangle. Since it is an altitude,
it forms two congruent triangles with angles measuring
,
, and
. So,
will be the product
of
(the shorter leg) and
.
yards, or approximately
yards.
Special Right Triangles in the Real World
You can use special right triangles in many real-world contexts. Many real-life applications of geometry rely
on special right triangles, so being able to recall and use these ratios is a way to save time when solving
problems.
Example 6
The diagram below shows the shadow a flagpole casts at a certain time of day.
If the length of the shadow cast by the flagpole is
m, what is the height of the flagpole and the length of the hypotenuse of the right triangle shown?
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The wording in this problem is complicated, but you only need to notice a few things. You can tell in the
picture that this triangle has angles of
,
, and
(This assumes that the flagpole is perpendicular
to the ground, but that is a safe assumption). The height of the flagpole is the longer leg in the triangle, so
use the special right triangle ratios to find the length of the hypotenuse.
The longer leg is the product of the shorter leg and
so the height of the flagpole is
. The length of the shorter leg is given as
meters,
m.
The length of the hypotenuse is the hypotenuse of a
length of the shorter leg, so it will equal
, or
meters.
triangle. It will always be twice the
Example 7
Antonio built a square patio in his backyard.
He wants to make a water pipe for flowers that goes from one corner to another, diagonally. How long will
that pipe be?
The first step in a word problem of this nature is to add important information to the drawing. Because the
problem asks you to find the length from one corner to another, you should draw that segment in.
Once you draw the diagonal path, you can see how triangles help answer this question. Because both legs
of the triangle have the same measurement (
feet), this is an isosceles right triangle. The angles in an
isosceles right triangle are
,
, and
.
In an isosceles right triangle, the hypotenuse is always equal to the product of the length of one leg and
. So, the length of Antonio’s water pipe will be the product of
and
, or
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feet. This value is approximately equal to
feet.
Lesson Summary
In this lesson, we explored how to work with different radicals both in theory and in practical situations.
Specifically, we have learned:
•
How to identify and use the ratios involved with right isosceles triangles.
•
How to identify and use the ratios involved with
•
How to identify and use ratios involved with equilateral triangles.
•
How to employ right triangle ratios when solving real-world problems.
triangles.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Lesson Exercises
1. Mildred had a piece of scrap wood cut into an equilateral triangle. She wants to cut it into two smaller
congruent triangles. What will be the angle measurement of the triangles that result?
2. Roberto has a square pizza. He wants to cut two congruent triangles out of the pizza without leaving any
leftovers. What will be the angle measurements of the triangles that result?
3. What is the length of the hypotenuse in the triangle below?
4. What is the length of the hypotenuse in the triangle below?
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5. What is the length of the longer leg in the triangle below?
6. What is the length of one of the legs in the triangle below?
7. What is the length of the shorter leg in the triangle below?
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8. A square window has a diagonal of
feet. What is the length of one of its sides?
9. A square block of foam is cut into two congruent wedges. If a side of the original block was
long is the diagonal cut?
feet, how
10. Thuy wants to find the area of an equilateral triangle but only knows that the length of one side is
inches. What is the height of Thuy’s triangle? What is the area of the triangle?
Answers
1.
,
, and
[Diff: 1]
2.
,
, and
[Diff: 1]
3.
[Diff: 2]
4.
cm or approx.
5.
miles or approx.
6.
7.
8.
9.
10.
cm [Diff: 2]
miles [Diff: 2]
mm [Diff: 2]
feet [Diff: 2]
feet [Diff: 3]
feet or approx.
inches or approx.
feet [Diff: 3]
in. The area is
Tangent Ratio
Learning Objectives
494
•
Identify the different parts of right triangles.
•
Identify and use the tangent ratio in a right triangle.
•
Identify complementary angles in right triangles.
2
inches [Diff: 3].
•
Understand tangent ratios in special right triangles.
Introduction
Now that you are familiar with right triangles, the ratios that relate the sides, as well as other important applications, it is time to learn about trigonometric ratios. Trigonometric ratios show the relationship between
the sides of a triangle and the angles inside of it. This lesson focuses on the tangent ratio.
Parts of a Triangle
In trigonometry, there are a number of different labels attributed to different sides of a right triangle. They
are usually in reference to a specific angle. The hypotenuse of a triangle is always the same, but the terms
adjacent and opposite depend on which angle you are referencing. A side adjacent to an angle is the leg
of the triangle that helps form the angle. A side opposite to an angle is the leg of the triangle that does not
help form the angle.
In the triangle shown above, segment
Similarly,
is adjacent to
is adjacent to
, and
is opposite
, and segment
is opposite to
. The hypotenuse is always
.
.
Example 1
Examine the triangle in the diagram below.
Identify which leg is adjacent to
, opposite to
, and the hypotenuse.
The first part of the question asks you to identify the leg adjacent to
. Since an adjacent leg is the one
that helps to form the angle and is not the hypotenuse, it must be
. The next part of the question asks
you to identify the leg opposite
. Since an opposite leg is the leg that does not help to form the angle,
it must be
. The hypotenuse is always opposite the right angle, so in this triangle the hypotenuse is
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segment
.
The Tangent Ratio
The first ratio to examine when studying right triangles is the tangent. The tangent of an angle is the ratio
of the length of the opposite side to the length of the adjacent side. The hypotenuse is not involved in the
tangent at all. Be sure when you find a tangent that you find the opposite and adjacent sides relative to the
angle in question.
For an acute angle measuring
, we define
.
Example 2
What are the tangents of
and
in the triangle below?
To find these ratios, first identify the sides opposite and adjacent to each angle.
So, the tangent of
It is common to write
is about
instead of
and the tangent of
is
.
. In this text we will use both notations.
Complementary Angles in Right Triangles
Recall that in all triangles, the sum of the measures of all angles must be
. Since a right angle has a
measure of
, the remaining two angles in a right triangle must be complementary. Complementary
angles have a sum of
. This means that if you know the measure of one of the smaller angles in a right
triangle, you can easily find the measure of the other. Subtract the known angle from
and you’ll have
the measure of the other angle.
Example 3
What is the measure of
in the triangle below?
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To find
, you can subtract the measure of
So, the measure of
is
since
and
from
.
are complementary.
Tangents of Special Right Triangles
It may help you to learn some of the most common values for tangent ratios. The table below shows you
values for angles in special right triangles.
Tangent
special right triangle. You can use these
Notice that you can derive these ratios from the
ratios to identify angles in a triangle. Work backwards from the ratio. If the ratio equals one of these values,
you can identify the measurement of the angle.
Example 4
What is
in the triangle below?
Find the tangent of
and compare it to the values in the table above.
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So, the tangent of
tangent of . So,
is
. If you look in the table, you can see that an angle that measures
.
has a
Example 5
What is
in the triangle below?
Find the tangent of
and compare it to the values in the table above.
So, the tangent of
has a tangent of
is about
. So,
Notice in this example that
. If you look in the table, you can see that an angle that measures
.
is a
triangle. You can use this fact to see that
.
Lesson Summary
In this lesson, we explored how to work with different radical expressions both in theory and in practical situations. Specifically, we have learned:
498
•
How to identify the different parts of right triangles.
•
How to identify and use the tangent ratio in a right triangle.
•
How to identify complementary angles in right triangles.
•
How to understand tangent ratios in special right triangles.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Lesson Exercises
Use the following diagram for exercises 1-5.
1. How long is the side opposite angle
2. How long is the side adjacent to angle
3. How long is the hypotenuse?
4. What is the tangent of
5. What is the tangent of
6. What is the measure of
in the diagram below?
7. What is the measure of
in the diagram below?
Use the following diagram for exercises 8-9.
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8. What is the tangent of
9. What is the tangent of
10. What is the measure of
Answers
1.
mm [Diff: 1]
2.
mm [Diff: 1]
3.
4.
mm [Diff: 1]
[Diff: 2]
5.
[Diff: 2]
6.
[Diff: 2]
7.
[Diff: 2]
8.
9.
10.
500
[Diff: 2]
[Diff: 2]
[Diff: 2]
in the triangle below?
Sine and Cosine Ratios
Learning Objectives
•
Review the different parts of right triangles.
•
Identify and use the sine ratio in a right triangle.
•
Identify and use the cosine ratio in a right triangle.
•
Understand sine and cosine ratios in special right triangles.
Introduction
Now that you have some experience with tangent ratios in right triangles, there are two other basic types of
trigonometric ratios to explore. The sine and cosine ratios relate opposite and adjacent sides of a triangle
to the hypotenuse. Using these three ratios and a calculator or a table of trigonometric ratios you can solve
a wide variety of problems!
Review: Parts of a Triangle
The sine and cosine ratios relate opposite and adjacent sides to the hypotenuse. You already learned these
terms in the previous lesson, but they are important to review and commit to memory. The hypotenuse of
a triangle is always opposite the right angle, but the terms adjacent and opposite depend on which angle
you are referencing. A side adjacent to an angle is the leg of the triangle that helps form the angle. A side
opposite to an angle is the leg of the triangle that does not help form the angle.
Example 1
Examine the triangle in the diagram below.
Identify which leg is adjacent to angle
hypotenuse.
, which leg is opposite to angle
The first part of the question asks you to identify the leg adjacent to
, and which segment is the
. Since an adjacent leg is the one
that helps to form the angle and is not the hypotenuse, it must be
. The next part of the question asks
you to identify the leg opposite
. Since an opposite leg is the leg that does not help to form the angle,
it must be
. The hypotenuse is always opposite the right angle, so in this triangle it is segment
.
The Sine Ratio
Another important trigonometric ratio is sine. A sine ratio must always refer to a particular angle in a right
triangle. The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Remember that in a ratio, you list the first item on top of the fraction and the second item on the
bottom.
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So, the ratio of the sine will be
.
Example 2
What are
and
in the triangle below?
All you have to do to find the solution is build the ratio carefully.
So,
and
.
The Cosine Ratio
The next ratio to examine is called the cosine. The cosine is the ratio of the adjacent side of an angle to
the hypotenuse. Use the same techniques you used to find sines to find cosines.
Example 3
What are the cosines of
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and
in the triangle below?
To find these ratios, identify the sides adjacent to each angle and the hypotenuse. Remember that an adjacent
side is the one that does create the angle and is not the hypotenuse.
So, the cosine of
is about
and the cosine of
is about
.
Note that
is NOT one of the special right triangles, but it is a right triangle whose sides are a
Pythagorean triple.
Sines and Cosines of Special Right Triangles
It may help you to learn some of the most common values for sine and cosine ratios. The table below shows
you values for angles in special right triangles.
Sine
Cosine
You can use these ratios to identify angles in a triangle. Work backwards from the ratio. If the ratio equals
one of these values, you can identify the measurement of the angle.
Example 4
What is the measure of
in the triangle below?
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Note: Figure is not to scale.
Find the sine of
and compare it to the values in the table above.
So, the sine of
sine of
. So,
is
. If you look in the table, you can see that an angle that measures
.
Example 5
What is the measure of
Find the cosine of
504
in the triangle below?
and compare it to the values in the previous table.
has a
So, the cosine of
is about
has a cosine of
. So,
. If you look in the table, you can see that an angle that measures
measures about
. This is a
right triangle.
Lesson Summary
In this lesson, we explored how to work with different trigonometric ratios both in theory and in practical situations. Specifically, we have learned:
•
The different parts of right triangles.
•
How to identify and use the sine ratio in a right triangle.
•
How to identify and use the cosine ratio in a right triangle.
•
How to apply sine and cosine ratios in special right triangles.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Points to Consider
Before you begin the next lesson, think about strategies you could use to simplify an equation that contains
a trigonometric function.
Note, you can only use the
,
, and
ratios on the acute angles of a right triangle. For now it only
makes sense to talk about the
,
, or
ratio of an acute angle. Later in your mathematics studies
you will redefine these ratios in a way that you can talk about
,
, and
of acute, obtuse, and
even negative angles.
Lesson Exercises
Use the following diagram for exercises 1-3.
1. What is the sine of
2. What is the cosine of
3. What is the cosine of
Use the following diagram for exercises 4-6.
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4. What is the sine of
5. What is the cosine of
6. What is the sine of
7. What is the measure of
in the diagram below?
Use the following diagram for exercises 8-9.
8. What is the sine of
9. What is the cosine of
10. What is the measure of
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in the triangle below?
Answers
1.
[Diff: 2]
2.
[Diff: 2]
3.
[Diff: 2]
4.
inches [Diff: 2]
5.
inches [Diff: 2]
6.
inches [Diff: 2]
7.
[Diff: 2]
8.
[Diff: 2]
9.
[Diff: 2]
10.
[Diff: 2]
507
Inverse Trigonometric Ratios
Learning Objectives
•
Identify and use the arctangent ratio in a right triangle.
•
Identify and use the arcsine ratio in a right triangle.
•
Identify and use the arccosine ratio in a right triangle.
•
Understand the general trends of trigonometric ratios.
Introduction
The word inverse is probably familiar to you—often in mathematics, after you learn to do an operation, you
also learn how to “undo” it. Doing the inverse of an operation is a way to undo the original operation. For
example, you may remember that addition and subtraction are considered inverse operations. Multiplication
and division are also inverse operations. In algebra you used inverse operations to solve equations and inequalities. You may also remember the term additive inverse, or a number that can be added to the original
to yield a sum of
. For example,
and
are additive inverses because
.
In this lesson you will learn to use the inverse operations of the trigonometric functions you have studied
thus far. You can use inverse trigonometric functions to find the measures of angles when you know the
lengths of the sides in a right triangle.
Inverse Tangent
When you find the inverse of a trigonometric function, you put the word arc in front of it. So, the inverse of
a tangent is called the arctangent (or arctan for short). Think of the arctangent as a tool you can use like
any other inverse operation when solving a problem. If tangent tells you the ratio of the lengths of the sides
opposite and adjacent to an angle, then arctan tells you the measure of an angle with a given ratio.
. The arctangent can be used to find the measure of
Suppose
equation.
on the left side of the
Where did that
come from? There are two basic ways to find an arctangent. Sometimes you will be
given a table of trigonometric values and the angles to which they correspond. In this scenario, find the value
that is closest to the one provided, and identify the corresponding angle.
Another, easier way of finding the arctangent is to use a calculator. The arctangent button may be labeled
“arctan,” “atan,” or “
.” Either way, select this button, and input the value in question. In this case,
you would press the arctangent button and enter
(or on some calculators, enter
, then press
“arctan”). The output will be the value of measure
.
is about
508
.
Example 1
Solve for
if
You can use the inverse of tangent, arctangent to find this value.
Then use your calculator to find the arctangent of
.
Example 2
What is
in the triangle below?
First identify the proper trigonometric ratio related to
that can be found using the sides given. The
tangent uses the opposite and adjacent sides, so it will be relevant here.
Now use the arctangent to solve for the measure of
Then use your calculator to find the arctangent of
.
.
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Inverse Sine
Just as you used arctangent as the inverse operation for tangent, you can also use arcsine (shortened as
arcsin) as the inverse operation for sine. The same rules apply. You can use it to isolate a variable for an
angle measurement, but you must perform the operation on both sides of the equation. When you know the
arcsine value, use a table or a calculator to find the measure of the angle.
Example 3
Solve for
if
You can use the inverse of sine, arcsine to find this value.
Then use your calculator to find the arcsine of
.
Example 4
What is
in the triangle below?
First identify the proper trigonometric ratio related to angle
that can be found using the sides given. The
sine uses the opposite side and the hypotenuse, so it will be relevant here.
Now use the arcsine to isolate the value of angle
Finally, use your calculator to find the arcsine of
510
.
.
Inverse Cosine
The last inverse trigonometric ratio is arccosine (often shortened to arccos). The same rules apply for arccosine as apply for all other inverse trigonometric functions. You can use it to isolate a variable for an angle
measurement, but you must perform the operation on both sides of the equation. When you know the arccosine value, use a table or a calculator to find the measure of the angle.
Example 5
Solve for
if
.
You can use the inverse of cosine, arccosine, to find this value.
Then use your calculator to find the arccosine of
.
Example 6
What is the measure of
in the triangle below?
First identify the proper trigonometric ratio related to
that can be found using the sides given. The
cosine uses the adjacent side and the hypotenuse, so it will be relevant here.
Now use the arccosine to isolate the value of
.
Finally use your calculator or a table to find the arccosine of
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General Trends in Trigonometric Ratios
Now that you know how to find the trigonometric ratios as well as their inverses, it is helpful to look at trends
in the different values. Remember that each ratio will have a constant value for a specific angle. In any right
triangle, the sine of a
angle will always be
—it doesn’t matter how long the sides are. You can use
that information to find missing lengths in triangles where you know the angles, or to identify the measure
of an angle if you know two of the sides.
Examine the table below for trends. It shows the sine, cosine, and tangent values for eight different angle
measures.
Sine
Cosine
Tangent
Example 7
Using the table above, which value would you expect to be greater: the sine of
or the cosine of
?
is
and the sine of
You can use the information in the table to solve this problem. The sine of
is
. So, the sine of
will be between the values
and
. The cosine of
is
and the cosine of
is
So, the cosine of
will be between the values of
and
Since the range for the cosine is greater, than the range for the sine, it can be assumed that the cosine of
will be greater than the sine of
Notice that as the angle measures approach
,
approaches . Similarly, as the value of the angles
approach
, the
approaches . In other words, as the
gets greater, the
gets smaller for
the angles in this table.
The tangent, on the other hand, increases rapidly from a small value to a large value (infinity, in fact) as the
angle approaches
.
Lesson Summary
In this lesson, we explored how to work with different radicals both in theory and in practical situations.
Specifically, we have learned:
512
•
How to identify and use the arctangent ratio in a right triangle.
•
How to identify and use the arcsine ratio in a right triangle.
•
How to identify and use the arccosine ratio in a right triangle.
•
How to understand the general trends of trigonometric ratios.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Points to Consider
To this point, all of the trigonometric ratios you have studied have dealt exclusively with right triangles. Can
you think of a way to use trigonometry on triangles that are acute or obtuse?
Lesson Exercises
1. Solve for
2. Solve for
3. What is the measure of
in the triangle below?
4. Solve for
5. What is the measure of
in the triangle below?
6. Solve for
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7. Solve for
8. What is the measure of
in the triangle below?
9. What is the measure of
in the triangle below?
10. What is the measure of
Answers
514
1.
[Diff: 1]
2.
[Diff: 1]
3.
[Diff: 2]
4.
[Diff: 1]
in the triangle below?
5.
[Diff: 3]
6.
[Diff: 2]
7.
[Diff: 2]
8.
[Diff: 3]
9.
[Diff: 3]
10.
[Diff: 3]
Acute and Obtuse Triangles
Learning Objectives
•
Identify and use the Law of Sines.
•
Identify and use the Law of Cosines.
Introduction
Trigonometry is most commonly learned on right triangles, but the ratios can have uses for other types of
triangles, too. This lesson focuses on how you can apply sine and cosine ratios to angles in acute or obtuse
. In an obtuse triangle,
triangles. Remember that in an acute triangle, all angles measure less than
there will be one angle that has a measure that is greater than
.
The Law of Sines
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of the angle opposite
it will be constant. That is, the ratio is the same for all three angles and their opposite sides. Thus, if you
find the ratio, you can use it to find missing angle measure and side lengths.
Note the convention that
denotes
and
is the length of the side opposite
.
Example 1
515
Examine the triangle in the following diagram.
What is the length of the side labeled j?
You can use the law of sines to solve this problem. Because you have one side and the angle opposite, you
can find the constant that applies to the entire triangle. This ratio will be equal to the proportion of side
and
. You can use your calculator to find the value of the sines.
So, using the law of sines, the length of
is about
Example 2
Examine the triangle in the following diagram.
What is the measure of
516
?
meters.
You can use the law of sines to solve this problem. Because you have one side and the angle opposite, you
can find the constant that applies to the entire triangle. This ratio will be equal to the proportion of side
and angle
. You can use your calculator to find the value of the sines.
So, using the law of sines, the angle labeled
must measure about
.
The Law of Cosines
There is another law that works on acute and obtuse triangles in addition to right triangles. The Law of
Cosines uses the cosine ratio to identify either lengths of sides or missing angles. To use the law of cosines,
you must have either the measures of all three sides, or the measure of two sides and the measure of the
included angle.
It doesn’t matter how you assign the variables to the three sides of the triangle, but the angle
opposite side .
must be
Example 3
Examine the triangle in the following diagram.
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What is the measure of side
Use the Law of Cosines to find
letter
.
So,
is about
?
. Since
is opposite
inches.
Example 4
Examine the triangle in the following diagram.
What is the measure of
Use the Law of Cosines to find the measure of
518
.
, we will call the length of
by the
So,
is about
.
Lesson Summary
In this lesson, we explored how to work with different radical expressions both in theory and in practical situations. Specifically, we have learned:
•
how to identify and use the law of sines.
•
how to identify and use the law of cosines.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Lesson Exercises
Exercises 1 and 2 use the triangle in the following diagram.
1. What is the length of side
2. What is
3. Examine the triangle in the following diagram.
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What is the measure of
4. Examine the triangle in the following diagram.
What is the measure of
5. Examine the triangle in the following diagram.
What is the measure of side
520
6. Examine the triangle in the following diagram.
What is the measure of
Use the triangle in the following diagram for exercises 7 and 8.
7. What is the measure of
8. What is the measure of
9. Examine the triangle in the following diagram.
What is the measure of
10. Examine the triangle in the following diagram.
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What is the measure of
Answers
1.
2.
inches [Diff: 3]
[Diff: 3]
3.
[Diff: 3]
4.
[Diff: 3]
5.
6.
7.
inches [Diff: 3]
[Diff: 3]
[Diff: 3]
8.
9.
10.
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[Diff: 3]
[Diff: 3]
9. Circles
About Circles
Learning Objectives
•
Distinguish between radius, diameter, chord, tangent, and secant of a circle.
•
Find relationships between congruent and similar circles.
•
Examine inscribed and circumscribed polygons.
•
Write the equation of a circle.
Circle, Center, Radius
A circle is defined as the set of all points that are the same distance away from a specific point called the
center of the circle. Note that the circle consists of only the curve but not of the area inside the curve. The
distance from the center to the circle is called the radius of the circle.
We often label the center with a capital letter and we refer to the circle by that letter. For example, the circle
below is called circle
or
.
Congruent Circles
Two circles are congruent if they have the same radius, regardless of where their centers are located. For
example, all circles of radius of centimeters are congruent to each other. Similarly, all circles with a radius
of
miles are congruent to each other. If circles are not congruent, then they are similar with the similarity
ratio given by the ratio of their radii.
Example 1
Determine which circles are congruent and which circles are similar. For similar circles find the similarity
ratio.
523
and
are congruent since they both have a radius of
and
are similar with similarity ratio of
.
and
are similar with similarity ratio of
.
and
are similar with similarity ratio of
.
and
are similar with similarity ratio of
.
and
are similar with similarity ratio of
.
cm.
Chord, Diameter, Secant
A chord is defined as a line segment starting at one point on the circle and ending at another point on the
circle.
A chord that goes through the center of the circle is called the diameter of the circle. Notice that the diameter
is twice as long as the radius of the circle.
A secant is a line that cuts through the circle and continues infinitely in both directions.
Point of Tangency and Tangent
A tangent line is defined as a line that touches the circle at exactly one point. This point is called the point
of tangency.
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Example 2
Identify the following as a secant, chord, diameter, radius, or tangent:
A.
B.
C.
D.
E.
F.
A.
is a diameter of the circle.
B.
is a radius of the circle.
C.
is a chord of the circle.
D.
is a tangent of the circle.
E.
is a secant of the circle.
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F.
is a radius of the circle.
Inscribed and Circumscribed Polygons
A convex polygon whose vertices all touch a circle is said to be an inscribed polygon. A convex polygon
whose sides all touch a circle is said to be a circumscribed polygon. The figures below show examples
of inscribed and circumscribed polygons.
Equations and Graphs of Circles
A circle is defined as the set of all points that are the same distance from a single point called the center.
This definition can be used to find an equation of a circle in the coordinate plane.
Let’s consider the circle shown below. As you can see, this circle has its center at point
radius of .
All the points
on the circle are a distance of
and it has a
units away from the center of the circle.
We can express this information as an equation with the help of the Pythagorean Theorem. The right triangle
shown in the figure has legs of length
and
and hypotenuse of length
We can generalize this equation for a circle with center at point (
526
,
. We write:
) and radius
.
Example 3
Find the center and radius of the following circles:
A.
B.
A. We rewrite the equation as:
the radius is .
The center of the circle is at point
and
The center of the circle is at point
B. We rewrite the equation as:
and the radius is .
Example 4
Graph the following circles:
A.
B.
In order to graph a circle, we first graph the center point and then draw points that are the length of the radius
away from the center.
A. We rewrite the equation as:
the radius is .
B. We rewrite the equation as:
and the radius is .
The center of the circle is point at
and
The center of the circle is point at
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Example 5
Write the equation of the circle in the graph.
From the graph we can see that the center of the circle is at point
and the radius is
Thus the equation is:
Example 6
Determine if the point
is on the circle given by the equation:
In order to find the answer, we simply plug the point
The point
into the equation of the circle.
satisfies the equation of the circle.
Example 7
Find the equation of the circle whose diameter extends from point
to
The general equation of a circle is:
528
units long.
In order to write the equation of the circle in this example, we need to find the center of the circle and the
radius of the circle.
Let’s graph the two points on the coordinate plane.
We see that the center of the circle must be in the middle of the diameter.
In other words, the center point is midway between the two points
and
. To get from point
to
point
, we must travel units to the right and units up. To get halfway from point
to point
, we
units to the right and
units up. This means the center of the circle is at point
must travel
or
.
We find the length of the radius using the Pythagorean Theorem:
Thus, the equation of the circle is:
Completing the Square:
You saw that the equation of a circle with center at point
) and radius
is given by:
This is called the standard form of the circle equation. The standard form is very useful because it tells us
right away what the center and the radius of the circle is.
If the equation of the circle is not in standard form, we use the method of completing the square to rewrite
the equation in the standard form.
Example 8
Find the center and radius of the following circle and sketch a graph of the circle.
To find the center and radius of the circle we need to rewrite the equation in standard form. The standard
equation has two perfect square factors one for the terms and one for the terms. We need to complete
the square for the
terms and the
terms separately.
529
To complete the squares we need to find which constants allow us to factors each trinomial into a perfect
square. To complete the square for the
terms we need to add a constant of on both sides.
To complete the square for the
terms we need to add a constant of
on both sides.
We can factor the separate trinomials and obtain:
This simplifies as:
You can see now that the center of the circle is at point
and the radius is
Concentric Circles
Concentric circles are circles of different radii that share the same center point.
Example 9
Write the equations of the concentric circles shown in the graph.
530
.
Example 10
Determine if the circles given by the equations
and
are concentric.
To find the answer to this question, we must rewrite the equations of the circles in standard form and find
the center point of each circle.
To rewrite in standard form, we complete the square on the
and
terms separately.
First circle:
The center of the first circle is also at point
so the circles are concentric.
Second circle:
The center of the second circle is at point
.
Lesson Summary
In this section we discussed many terms associated with circles and looked at inscribed and circumscribed
polygons. We also covered graphing circles on the coordinate grid and finding the equation of a circle. We
found that sometimes we need to use the technique of completing the square to find the equation of a circle.
Lesson Exercises
1. Identify each of the following as a diameter, a chord, a radius, a tangent, or a secant line.
531
a.
b.
c.
d.
e.
f.
2. Determine which of the following circles are congruent and which are similar. For circles that are similar
give the similarity ratio.
For exercises 3-8, find the center and the radius of the circles:
3.
532
4.
5.
6.
7.
8.
9. Check that the point
10. Check that the point
is on the circle given by the equation
is on the circle given by the equation
11. Write the equation of the circle with center at
and radius
.
12. Write the equation of the circle with center at
and radius
.
13. Write the equation of the circle with center at
and radius
.
For 14 and 15, write the equation of the circles.
14.
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15.
16. In a circle with center
ameter.
one endpoint of a diameter is
17. The endpoints of the diameter of a circle are given by the points
equation of the circle.
18. A circle has center
and contains point
19. A circle has center
21. Find the center and the radius of the following circle:
22. Find the center and the radius of the following circle:
23. Determine if the circles given by the equations are concentric.
b.
c.
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and
and
. Find the
. Find the equation of the circle.
20. Find the center and the radius of the following circle:
and
and
. Find the equation of the circle.
and contains point
a.
. Find the other endpoint of the di-
.
.
.
d.
and
Answers
1.
a.
is a radius.
b.
is a diameter.
c.
is a tangent.
d.
is a secant.
e.
is a chord.
f.
is a radius.
2.
is congruent to
with similarity ratio
.
3. The center is located at
4. The center is located at
5. The center is located at
6. The center is located at
7. The center is located at
8. The center is located at
9.
10.
;
is similar to
, radius =
is similar to
.
.
, radius =
, radius =
;
.
, radius =
, radius =
with similarity ratio
.
.
, radius =
.
The point is on the circle.
The point is not on the circle.
11.
12.
13.
14.
15.
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16.
17.
18.
19.
20. The center is located at
, radius =
.
21. The center is located at
, radius =
.
22. The center is located at
, radius =
.
23.
a. Yes
b. No
c. No
d. Yes
Tangent Lines
Learning Objectives
•
Find the relationship between a radius and a tangent to a circle.
•
Find the relationship between two tangents drawn from the same point.
•
Circumscribe a circle.
•
Find equations of concentric circles.
Introduction
In this section we will discuss several theorems about tangent lines to circles and the applications of these
theorems to geometry problems. Remember that a tangent to a circle is a line that intersects the circle at
exactly one point and that this intersection point is called the point of tangency.
536
Tangent to a Circle
Tangent to a Circle Theorem: A tangent line is always at right angles to the radius of
the circle at the point of tangency.
Proof. We will prove this theorem by contradiction.
We start by making a drawing.
is a radius of the circle.
point of intersection between the radius and the tangent line.
is the center of the circle and
is the
Assume that the tangent line is not perpendicular to the radius.
537
There must be another point
Therefore, in the right triangle
on the tangent line such that
,
this is not possible because
is the hypotenuse and
. (Note that
is perpendicular to the tangent line.
is a leg of the triangle. However,
length of the radius
).
Since our assumption led us to a contradiction, this means that our assumption was incorrect. Therefore,
the tangent line must be perpendicular to the radius of the circle.
Since the tangent to a circle and the radius of the circle make a right angle with each other, we can often
use the Pythagorean Theorem in order to find the length of missing line segments.
Example 1
In the figure,
is tangent to the circle. Find
.
Since
is tangent to the circle, then
This means that
of
is a right triangle and we can apply the Pythagorean Theorem to find the length
.
Example 2
Mark is standing at the top of Mt. Whitney, which is
feet tall. The radius of the Earth is approximately
miles. (There are
538
.
feet in one mile.) How far can Mark see to the horizon?
We start by drawing the figure above.
The distance to the horizon is given by the line segment
.
Let us convert the height of the mountain from feet into miles.
Since
is tangent to the Earth,
is a right triangle and we can use the Pythagorean Theorem.
Converse of a Tangent to a Circle
Converse of a Tangent to a Circle Theorem If a line is perpendicular to the radius of
a circle at its outer endpoint, then the line is tangent to the circle.
539
Proof.
We will prove this theorem by contradiction. Since the line is perpendicular to the radius at its outer endpoint
it must touch the circle at point
. For this line to be tangent to the circle, it must only touch the circle at
this point and no other.
Assume that the line also intersects the circle at point
Since both
and
.
are radii of the circle,
, and
is isosceles.
.
This means that
It is impossible to have two right angles in the same triangle.
We arrived at a contradiction so our assumption must be incorrect. We conclude that line
to the circle at point
.
Example 3
Determine whether
is tangent to the circle.
is tangent to the circle if
To show that
540
.
is a right triangle we use the Converse of the Pythagorean Theorem:
is tangent
The lengths of the sides of the triangle satisfy the Pythagorean Theorem, so
and is therefore tangent to the circle.
is perpendicular to
Tangent Segments from a Common External Point
Tangent Segments from a Common External Point Theorem If two segments from
the same exterior point are tangent to the circle, then they are congruent.
Proof.
The figure above shows a diagram of the situation.
•
Given:
•
Prove:
is a tangent to the circle and
is a tangent to the circle
Statement
Reason
1.
1. Given
is tangent to the circle
2. Tangent to a Circle Theorem
2.
3.
is a tangent to the circle
3. Given
4.
4. Tangent to a Circle Theorem
5.
5. Radii of same the circle
6.
6. Same line
7.
8.
7. Hypotenuse-Leg congruence
8. Congruent Parts of Congruent Triangles are Congruent
541
Example 4
Find the perimeter of the triangle.
All sides of the triangle are tangent to the circle.
The Tangent Segments from a Common External Point Theorem tells us that:
The perimeter of the triangle =
=
Example 5
An isosceles right triangle is circumscribed about a circle with diameter of
of the triangle.
inches. Find the hypotenuse
Let’s start by making a sketch.
Since
and
are radii of the circle and
and
Therefore, quadrilateral
and
are tangents to the circle,
.
is a square.
Therefore,
We can find the length of side
542
by using the Pythagorean Theorem.
and
are both isosceles right triangles, therefore
corresponding sides are proportional:
We can find the length of
and all the
by using one of the ratios above:
Cross-multiply to obtain:
Rationalize the denominators:
The length of the hypotenuse is
+
Corollary to Tangent Segments Theorem
A line segment from an external point to the center of a circle bisects the angle formed by the tangent segments starting at that same external point.
Proof.
•
Given:
543
•
is a tangent to the circle
•
is a tangent to the circle
•
•
is the center of the circle
Prove
•
Proof.
We will use a similar figure to the one we used we used to prove the tangent segments theorem (pictured
above).
by
Therefore,
congruence.
.
Example 6
Show that the line
is tangent to the circle
. Find an equation for the line perpendicular to the tangent line at the point of tangency. Show that this line
goes through the center of the circle.
To check that the line is tangent to the circle, substitute the equation of the line into the equation for the
circle.
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This has a double root at
=
. This means that the line intersects the circle at only one point
.
A perpendicular line to the tangent line would have a slope that is the negative reciprocal of the tangent
line or
=
.
The equation of the line can be written:
We find the value of
The equation is
=
+
.
by plugging in the tangency point:
and we know that it passes through the origin since the
-intercept is zero.
This means that the radius of the circle is perpendicular to the tangent to the circle.
Lesson Summary
In this section we learned about tangents and their relationship to the circle. We found that a tangent line
touches the circle at one point, which is the endpoint of a radius of the circle. The radius and tangent line
are perpendicular to each other. We found out that if two segments are tangent to a circle, and share a
common endpoint outside the circle; the segments are congruent.
Lesson Exercises
1. Determine whether each segment is tangent to the given circle:
a.
b.
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c.
2. Find the measure of angle
a.
b.
c.
546
.
3. Find the missing length:
a.
b.
c.
547
4. Find the values of the missing variables
a.
b.
c.
548
5. Find the perimeter of the pentagon:
6. Find the perimeter of the parallelogram:
7. Find the perimeter of the right triangle:
8. Find the perimeter of the polygon:
549
9. Draw the line
point.
=
+
and the circle
+
=
. Show that these graphs touch at only one
Find the slope of the segment that joins this point to the center of the circle, and compare your answer with
the slope of the line
=
+
Answers
1.
a. Yes
b. Yes
c. No
2.
a.
b.
c.
3.
a.
b.
c.
4.
a.
b.
c.
5.
550
.
6.
7.
8.
9.
solve for
The slope of the line from
line.
to obtain double root
to
.
, which is the negative reciprocal of the slope of the
This means that the tangent line and radius are perpendicular.
Common Tangents and Tangent Circles
Learning Objectives
•
Solve problems involving common internal tangents of circles.
•
Solve problems involving common external tangents of circles.
•
Solve problems involving externally tangent circles.
•
Solve problems involving internally tangent circles.
Common tangents to two circles may be internal or external. A common internal tangent intersects the
line segment connecting the centers of the two circles whereas a common external tangent does not.
Common External Tangents
Here is an example in which you might encounter the use of common external tangents.
Example 1
Find the distance between the centers of the circles in the figure.
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Let’s label the diagram and draw a line segment that joins the centers of the two circles. Also draw the
segment
Since
perpendicular the radius
is tangent to both circles,
We can see that
This means that
is a rectangle, therefore
in
is a right triangle with
to find the missing side,
is perpendicular to both radii:
in
and
.
in.
in.
in and
in. We can apply the Pythagorean Theorem
.
The distance between the centers is approximately
inches.
Common Internal Tangents
Here is an example in which you might encounter the use of common internal tangents.
Example 2
is tangent to both circles. Find the value of
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and the distance between the centers of the circles.
Tangent is perpendicular to the radius
Tangent is perpendicular to the radius
Both equal
Vertical angles
similarity postulate
Therefore,
Using the Pythagorean Theorem on
Using the Pythagorean Theorem on
The distance between the centers of the circles is
Two circles are tangent to each other if they have only one common point. Two circles that have two common
points are said to intersect each other.
Two circles can be externally tangent if the circles are situated outside one another and internally tangent
if one of them is situated inside the other.
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Externally Tangent Circles
Here are some examples involving externally tangent circles.
Example 3
Circles tangent at
are centered at
and
. Line
is tangent to both circles at
. Find the radius of the smaller circle if
.
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tangent is perpendicular to the radius.
tangent is perpendicular to the radius.
In the right triangle
We are also given that
.
.
Therefore,
Also,
Therefore,
by the
similarity postulate.
The radius of the smaller circle is approximately
.
Example 4
Two circles that are externally tangent have radii of
inches and
inches respectively. Find the length of tangent
555
.
Label the figure as shown.
In
,
and
.
Therefore,
tangent is perpendicular to the radius.
tangent is perpendicular to the radius.
Therefore,
both equal
same angle.
~
by the
similarity postulate.
Therefore,
Internally Tangent Circles
Here is an example involving internally tangent circles.
Example 5
Two diameters of a circle of radius 15 inches are drawn to make a central angle of
. A smaller circle
is placed inside the bigger circle so that it is tangent to the bigger circle and to both diameters. What is the
556
radius of the smaller circle?
and
bisects
In
Draw
In
are two tangents to the smaller circle from a common point so by Theorem 9-3, \overline{ON}
we use
from the points of tangency between the circles perpendicular to
.
we use
We also have
because a tangent is perpendicular to the radius.
Therefore,
both equal
same angle.
Therefore,
by the
similarity postulate.
This gives us the ratio
557
(
since they are both radii of the small circle).
Lesson Summary
In this section we learned about externally and internally tangent circles. We looked at the different cases
when two circles are both tangent to the same line, and/or tangent to each other.
Lesson Exercises
is tangent to both circles.
1.
. Find
.
2.
and
. Find
.
3.
and
. Find
.
4.
and
Find
.
is tangent to both circles. Find the measure of angle
5.
6.
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and
and
.
7.
8.
For 9 and 10, find
9.
10.
Circles tangent at
.
;
;
are centered at
the smaller circle if
11.
and
.
is tangent to both circles at
. Find the radius of
.
,
12.
13. Four identical coins are lined up in a row as shown. The distance between the centers of the first and
the fourth coin is
inches. What is the radius of one of the coins?
559
14. Four circles are arranged inside an equilateral triangle as shown. If the triangle has sides equal to
cm, what is the radius of the bigger circle? What are the radii of the smaller circles?
15. In the following drawing, each segment is tangent to each circle. The largest circle has a radius of
inches. The medium circle has a radius of inches. What is the radius of the smallest circle tangent to the
medium circle?
16. Circles centered at
17.
and
are tangent at
. Show that
is a common external tangent to the two circles.
,
and
are collinear.
is tangent to both circles. Prove that
18. A circle with a -inch radius is centered at
, and a circle with a
-inch radius is centered at
,
where
and
are
inches apart. The common external tangent touches the small circle at
and
the large circle at
Answers
1.
2.
3.
4.
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. What kind of quadrilateral is
What are the lengths of its sides?
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16. Proof
17. Proof
18. Right trapezoid;
Arc Measures
Learning Objectives
•
Measure central angles and arcs of circles.
•
Find relationships between adjacent arcs.
•
Find relationships between arcs and chords.
Arc, Central Angle
In a circle, the central angle is formed by two radii of the circle with its vertex at the center of the circle. An
arc is a section of the circle.
561
Minor and Major Arcs, Semicircle
A semicircle is half a circle. A major arc is longer than a semicircle and a minor arc is shorter than a
semicircle.
An arc can be measured in degrees or in a linear measure (cm, ft, etc.). In this lesson we will concentrate
on degree measure. The measure of the minor arc is the same as the measure of the central angle that
corresponds to it. The measure of the major arc equals to
minus the measure of the minor arc.
Minor arcs are named with two letters—the letters that denote the endpoints of the arc. In the figure above,
the minor arc corresponding to the central angle
is called
. In order to prevent confusion,
major arcs are named with three letters—the letters that denote the endpoints of the arc and any other point
on the major arc. In the figure, the major arc corresponding to the central angle
.
is called
Congruent Arcs
Two arcs that correspond to congruent central angles will also be congruent. In the figure below,
because they are vertical angles. This also means that
.
Arc Addition Postulate
The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.
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In other words,
=
+
.
Congruent Chords Have Congruent Minor Arcs
In the same circle or congruent circles, congruent chords have congruent minor arcs.
Proof. Draw the following diagram, in which the chords
Construct
tively.
and
Then,
and
by drawing the radii for the center
by the
are congruent.
to points
and
respec-
postulate.
This means that central angles,
, which leads to the conclusion that
.
Congruent Minor Arcs Have Congruent Chords and Congruent Central Angles
In the same circle or congruent circles, congruent chords have congruent minor arcs.
Proof. Draw the following diagram, in which the
. In the diagram
,
,
, and
are radii of the circle.
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Since
, this means that the corresponding central angles are also congruent:
.
Therefore,
by the
We conclude that
postulate.
.
Here are some examples in which we apply the concepts and theorems we discussed in this section.
Example 1
Find the measure of each arc.
A.
B.
C.
A.
B.
C.
Example 2
Find
in circle
. The measures of all three arcs must add to
.
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Example 3
The circle
goes through
and
. Find
.
Draw the radii to points
and
.
We know that the measure of the minor arc
is equal to the measure of the central angle.
565
Lesson Summary
In this section we learned about arcs and chords, and some relationships between them. We found out that
there are major and minor arcs. We also learned that if two chords are congruent, so are the arcs they intersect, and vice versa.
Lesson Exercises
1. In the circle
identify the following:
a. four radii
b. a diameter
c. two semicircles
d. three minor arcs
e. two major arcs
2. Find the measure of each angle in
a.
566
:
b.
c.
d.
e.
f.
3. Find the measure of each angle in
:
a.
b.
c.
d.
e.
f.
4. The students in a geometry class were asked what their favorite pie is. The table below shows the result
of the survey. Make a pie chart of this information, showing the measure of the central angle for each slice
of the pie.
Kind of pie
Number of students
apple
pumpkin
cherry
lemon
chicken
banana
total
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5. Three identical pipes of diameter
inches are tied together by a metal band as shown. Find the length
of the band surrounding the three pipes.
6. Four identical pipes of diameter
inches are tied together by a metal band as shown. Find the length
of the band surrounding the four pipes.
Answers
1.
a.
b.
c.
,
,
d. Some possibilities:
e. Some possibilities:
2.
a.
b.
c.
d.
e.
f.
568
,
,
,
,
,
,
.
.
3.
a.
b.
c.
d.
e.
f.
4.
5.
in.
6.
in.
Chords
Learning Objectives
•
Find the lengths of chords in a circle.
•
Find the measure of arcs in a circle.
Introduction
Chords are line segments whose endpoints are both on a circle. The figure shows an arc
chord
and its related
.
569
There are several theorems that relate to chords of a circle that we will discuss in the following sections.
Perpendicular Bisector of a Chord
Theorem The perpendicular bisector of a chord is a diameter.
Proof
We will draw two chords,
and
such that
for any point
We can see that
The congruence of the triangles can be proven by the
and
This means that
.
postulate:
.
and
lies along
the center of the circle is one such point, it must lie along
is the midpoint of
on chord
then
and
, by the perpendicular bisector theorem. Since
so
is a diameter.
are radii of the circle and
is a diameter of the circle.
Perpendicular Bisector of a Chord Bisects Intercepted Arc
Theorem The perpendicular bisector of a chord bisects the arc intercepted by the chord.
Proof
570
.
are right angles
Any point that is equidistant from
If
is the perpendicular bisector of
We can see that
and
because of the
postulate.
are right angles.
This means that
.
This completes the proof.
Congruent Chords Equidistant from Center
Theorem Congruent chords in the same circle are equidistant from the center of the circle.
First, recall that the definition of distance from a point to a line is the length of the perpendicular segment
drawn to the line from the point.
and
are these distances, and we must prove that they are equal.
Proof.
by the SSS Postulate.
(given)
(radii)
(radii)
Since the triangles are congruent, their corresponding altitudes
and
must also be congruent:
.
Therefore,
and are equidistant from the center.
Converse of Congruent Chords Theorem
Theorem Two chords equidistant from the center of a circle are congruent.
This proof is left as a homework exercise.
Next, we will solve a few examples that apply the theorems we discussed.
Example 1
571
=
inches, and is
in. from the center of circle
.
A. Find the radius of the circle.
B. Find
Draw the radius
A.
.
is the hypotenuse of the right triangle
in.;
.
in.
Apply the Pythagorean Theorem.
B. Extend
to intersect the circle at point
Example 2
Two concentric circles have radii of
572
.
inches and
inches. A segment tangent to the smaller circle is a chord of the larger circle. What is the length of the segment?
Start by drawing a figure that represents the problem.
in.
in.
is a right triangle because the radius
at point
of the smaller circle is perpendicular to the tangent
.
Apply the Pythagorean Theorem.
Example 3
Find the length of the chord of the circle.
that is given by line
.
First draw a graph that represents the problem.
573
Find the intersection point of the circle and the line by substituting for
in the circle equation.
Solve using the quadratic formula.
or
The corresponding values of
are
or
Thus, the intersection points are approximately
and
.
We can find the length of the chord by applying the distance formula:
units.
Example 4
Let
and
be the positive
. Let
the circle
and
be the positive
574
-intercept, respectively, of the circle
-intercept and the positive
-intercept, respectively, of
.
Verify that the ratio of chords
For the circle
-intercept and the positive
is the same as the ratio of the corresponding diameters.
, the
-intercept is found by setting
. So
.
The
-intercept is found by setting
. So,
.
can be found using the distance formula:
For the circle
,
and
.
.
The ratio of the
.
Diameter of circle
is
Diameter of circle
is
.
.
The ratio of the diameters is
The ratio of the chords and the ratio of the diameters are the same.
Lesson Summary
In this section we gained more tools to find the length of chords and the measure of arcs. We learned that
the perpendicular bisector of a chord is a diameter and that the perpendicular bisector of a chord also bisects
the corresponding arc. We found that congruent chords are equidistant from the center, and chords
equidistant from the center are congruent.
Lesson Exercises
1. Find the value of
:
a.
b.
575
c.
d.
e.
f.
576
g.
h.
2. Find the measure of
.
a.
577
b.
c.
d.
e.
578
f.
g.
h.
579
3. Two concentric circles have radii of inches and inches. A segment tangent to the smaller circle is a
chord of the larger circle. What is the length of the segment?
4. Two congruent circles intersect at points
and
connecting the centers of the two circles measures
5. Find the length of the chord of the circle
.
is a chord to both circles. If the segment
in and
=
in, how long is the radius?
that is given by line
.
6. Prove Theorem 9-9.
7. Sketch the circle whose equation is
. Using the same system of coordinate axes, graph
, which should intersect the circle twice—at
the line
second quadrant. Find the coordinates of
.
8. Also find the coordinates for a point
and at another point
on the circle above, such that
.
9. The line
intersects the circle
in two points. Call the third quadrant point
, and find their coordinates. Let
be the point where the line through
the first quadrant point
the center of the circle intersects the circle again. Show that
1.
a.
b.
c.
d.
580
and
and
is a right triangle.
10. A circular playing field
meters in diameter has a straight path cutting across it. It is
the center of the field to the closest point on this path. How long is the path?
Answers
in the
meters from
e.
f.
g.
h.
2.
a.
b.
c.
d.
e.
f.
g.
h.
3.
4.
5.
6. proof
7.
8.
581
9.
;
;
10.
;
;
;
meters
Inscribed Angles
Learning Objective
•
Find the measure of inscribed angles and the arcs they intercept
Inscribed Angle, Intercepted Arc
An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle.
An inscribed angle is said to intercept an arc of the circle. We will prove shortly that the measure of an inscribed angle is half of the measure of the arc it intercepts.
Notice that the vertex of the inscribed angle can be anywhere on the circumference of the circle--it does not
need to be diametrically opposite the intercepted arc.
Measure of Inscribed Angle
The measure of a central angle is twice the measure of the inscribed angle that intercepts the same arc.
Proof.
582
and
both intercept
is a central angle and angle
is an inscribed
angle.
We draw the diameter of the circle through points
We see that
is isosceles because
and
, and let
and
and
are radii of the circle and are therefore congruent.
From this we can conclude that
Similarly, we can conclude that
We use the property that the sum of angles inside a triangle equals
and
to find that:
.
Then,
and
Therefore
.
Inscribed Angle Corollaries a-d
The theorem above has several corollaries, which will be left to the student to prove.
a. Inscribed angles intercepting the same arc are congruent
b. Opposite angles of an inscribed quadrilateral are supplementary
c. An angle inscribed in a semicircle is a right angle
d. An inscribed right angle intercepts a semicircle
Here are some examples the make use of the theorems presented in this section.
Example 1
Find the angle marked
in the circle.
583
The
is twice the measure of the angle at the circumference because it is a central angle.
Therefore,
This means that
Example 2
Find the angles marked
in the circle.
So,
Example 3
Find the angles marked
and
584
in the circle.
First we use
to find the measure of angle
Therefore,
.
.
because they are inscribed angles and intercept the same arc
.
In
,
.
Lesson Summary
In this section we learned about inscribed angles. We found that an inscribed angle is half the measure of
the arc it intercepts. We also learned some corollaries related to inscribed angles and found that if two inscribed angles intercept the same arc, they are congruent.
Lesson Exercises
1. In
,
,
and
. Find the measure of each angle:
a.
b.
c.
d.
585
e.
f.
2. Quadrilateral
is inscribed in
such that
,
,
.
Find the measure of each of the following angles:
a.
b.
c.
d.
3. In the following figure,
Find the following measures:
a.
b.
c.
d.
4. Prove the inscribed angle theorem corollary a.
5. Prove the inscribed angle theorem corollary b.
586
and
.
6. Prove the inscribed angle theorem corollary c.
7. Prove the inscribed angle theorem corollary d.
8. Find the measure of angle
.
a.
b.
c.
d.
587
9. Find the measure of the angles
and
.
a.
b.
10. Suppose that
the line through
the midpoint of
Answers
1.
a.
b.
588
is a diameter of a circle centered at
that is parallel to
.
, and let
, and
is any other point on the circle. Draw
be the point where it meets
. Prove that
is
c.
d.
e.
f.
2.
a.
b.
c.
d.
3.
a.
b.
c.
d.
4. Proof
5. Proof
6. Proof
7. Proof
8.
a.
b.
c.
d.
9.
a.
b.
10. Hint:
, so
.
589
Angles of Chords, Secants, and Tangents
Learning Objective
•
find the measures of angles formed by chords, secants, and tangents
Measure of Tangent-Chord Angle
Theorem The measure of an angle formed by a chord and a tangent that intersect on the circle equals
half the measure of the intercepted arc.
In other words:
and
Proof
Draw the radii of the circle to points
and
.
is isosceles, therefore
=
We also know that,
590
because
is tangent to the circle.
We obtain
Since
is a central angle that corresponds to
then,
.
This completes the proof.
Example 1
Find the values of
and
.
First we find angle
Using the Measure of the Tangent Chord Theorem we conclude that:
and
Therefore,
Angles Inside a Circle
Theorem The measure of the angle formed by two chords that intersect inside a circle is equal to half the
sum of the measure of their intercepted arcs. In other words, the measure of the angle is the average (mean)
of the measures of the intercepted arcs.
591
In this figure,
Proof
Draw a segment to connect points
and
.
Inscribed angle
Inscribed angle
+
+
Example 2
Find
.
592
The measure of an exterior angle in a triangle is equal
to the sum of the measures of the remote interior angles.
Substitution
Angles Outside a Circle
Theorem The measure of an angle formed by two secants drawn from a point outside the circle is equal to
half the difference of the measures of the intercepted arcs.
In other words:
This theorem also applies for an angle formed by two tangents to the circle drawn from a point outside the
circle and for an angle formed by a tangent and a secant drawn from a point outside the circle.
Proof
593
Draw a line to connect points
and
.
Inscribed angle
Inscribed angle
The measure of an exterior angle in a triangle is equal
to the sum of the measures of the remote interior angles.
Substitution
Example 3
Find the measure of angle
.
Lesson Summary
In this section we learned about finding the measure of angles formed by chords, secants, and tangents.
We looked at the relationship between the arc measure and the angles formed by chords, secants, and
594
tangents.
Lesson Exercises
1. Find the value of the variable.
a.
b.
c.
d.
595
e.
f.
g.
h.
596
i.
j.
k.
597
l.
m.
n.
598
o.
p.
q.
r.
599
2. Find the measure of the following angles:
a.
b.
c.
d.
e.
f.
3. Find the measure of the following angles:
a.
b.
c.
d.
4. Four points on a circle divide it into four arcs, whose sizes are
,
,
, and
, in consecutive order. The four points determine two intersecting chords. Find the sizes of the angles formed by
600
the intersecting chords.
Answers
1.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o.
p.
q.
r.
2.
a.
b.
c.
d.
601
e.
f.
3.
a.
b.
c.
d.
4.
and
Segments of Chords, Secants, and Tangents
Learning Objectives
•
Find the lengths of segments associated with circles.
In this section we will discuss segments associated with circles and the angles formed by these segments.
The figures below give the names of segments associated with circles.
Segments of Chords
Theorem If two chords intersect inside the circle so that one is divided into segments of length a and b and
the other into segments of length b and c then the segments of the chords satisfy the following relationship:
This means that the product of the segments of one chord equals the product of segments of the second
chord.
602
Proof
We connect points
and
and points
and
to make
and
.
Vertical angles
Inscribed angles intercepting the same arc
Inscribed angles intercepting the same arc
Therefore,
by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
Example 1
Find the value of the variable.
603
Segments of Secants
Theorem If two secants are drawn from a common point outside a circle and the segments are labeled as
below, then the segments of the secants satisfy the following relationship:
This means that the product of the outside segment of one secant and its whole length equals the product
of the outside segment of the other secant and its whole length.
Proof
We connect points
and
and points
and
to make
and
.
Same angle
Inscribed angles intercepting the same arc
Therefore,
by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
604
Example 2
Find the value of the variable.
Segments of Secants and Tangents
Theorem If a tangent and a secant are drawn from a point outside the circle then the segments of the secant
and the tangent satisfy the following relationship
This means that the product of the outside segment of the secant and its whole length equals the square
of the tangent segment.
Proof
We connect points
and
and points
and
to make
and
.
605
The measure of an Angle outside a circle is
equal to half the difference of the measures
of the intercepted arcs
The measure of an exterior angle in a triangle equals the sum of the measures of the
remote interior angles
Combining the two steps above
algebra
Therefore,
by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
Example 3
Find the value of the variable
assuming that it represents the length of a tangent segment.
Lesson Summary
In this section, we learned how to find the lengths of different segments associated with circles: chords,
secants, and tangents. We looked at cases in which the segments intersect inside the circle, outside the
circle, or where one is tangent to the circle. There are different equations to find the segment lengths, relating
606
to different situations.
Lesson Exercises
1. Find the value of missing variables in the following figures:
a.
b.
c.
d.
607
e.
f.
g.
608
h.
i.
j.
k.
609
l.
m.
n.
610
o.
p.
q.
r.
611
s.
t.
2. A circle goes through the points
at
612
. Given that
and
and
consecutively. The chords
find
?
and
intersect
3. Suzie found a piece of a broken plate. She places a ruler across two points on the rim, and the length of
the chord is found to be inches. The distance from the midpoint of this chord to the nearest point on the
rim is found to be inch. Find the diameter of the plate.
4. Chords
and
intersect at
. Given
and
find
.
Answers
1.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
or
o.
p.
q.
r.
s.
t.
2.
3.
inches.
613
4.
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10. Perimeter and Area
Triangles and Parallelograms
Learning Objectives
•
Understand basic concepts of the meaning of area.
•
Use formulas to find the area of specific types of polygons.
Introduction
Measurement is not a new topic. You have been measuring things nearly all your life. Sometimes you use
standard units (pound, centimeter), sometimes nonstandard units (your pace or arm span). Space is measured
according to its dimension.
•
One-dimensional space: measure the length of a segment on a line.
•
Two-dimensional space: measure the area that a figure takes up on a plane (flat surface).
•
Three-dimensional space: measure the volume that a solid object takes up in “space.”
In this lesson, we will focus on basic ideas about area in two-dimensional space. Once these basic ideas
are established we’ll look at the area formulas for some of the most familiar two-dimensional figures.
Basic Ideas of Area
Measuring area is just like measuring anything; before we can do it, we need to agree on standard units.
People need to say, “These are the basic units of area.” This is a matter of history. Let’s re-create some of
the thinking that went into decisions about standard units of area.
Example 1
What is the area of the rectangle below?
What should we use for a basic unit of area?
As one possibility, suppose we decided to use the space inside this circle as the unit of area.
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To find the area, you need to count how many of these circles fit into the rectangle, including parts of circles.
So far you can see that the rectangle’s space is made up of whole circles. Determining the fractional
parts of circles that would cover the remaining white space inside the rectangle would be no easy job! And
this is just for a very simple rectangle. The challenge is even more difficult for more complex shapes.
Instead of filling space with circles, people long ago realized that it is much simpler to use a square shape
for a unit of area. Squares fit together nicely and fill space with no gaps. The square below measures foot
on each side, and it is called square foot.
Now it’s an easy job to find the area of our rectangle.
The area is square feet, because
cover, the rectangle.
is the number of units of area (square feet) that will exactly fill, or
The principle we used in Example 1 is more general.
The area of a two-dimensional figure is the number of square units that will fill, or cover, the figure.
Two Area Postulates
Congruent Areas If two figures are congruent, they have the same area.
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This is obvious because congruent figures have the same amount of space inside them. However, two figures
with the same area are not necessarily congruent.
Area of Whole is Sum of Parts If a figure is composed of two or more parts that do not
overlap each other, then the area of the figure is the sum of the areas of the parts.
This is the familiar idea that a whole is the sum of its parts. In practical problems you may find it helpful to
break a figure down into parts.
Example 2
Find the area of the figure below.
Luckily, you don’t have to learn a special formula for an irregular pentagon, which this figure is. Instead,
you can break the figure down into a trapezoid and a triangle, and use the area formulas for those figures.
Basic Area Formulas
Look back at Example 1 and the way it was filled with unit area squares.
Notice that the dimensions are:
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base (or length)
feet
height (or width)
feet
But notice, too, that the base is the number of feet in one row of unit squares, and the height is the number
of rows. A counting principle tells us that the total number of square feet is the number in one row multiplied
by the number of rows.
Area
Area of a Rectangle If a rectangle has base
units.
Example 3
What is the area of the figure shown below?
Break the figure down into two rectangles.
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units and height
units, then the area,
, is
square
Area
Now we can build on the rectangle formula to find areas of other shapes.
Parallelogram
Example 4
How could we find the area of this parallelogram?
Make it into a rectangle
The rectangle is made of the same parts as the parallelogram, so their areas are the same. The area of
, so the area of the parallelogram is also
.
the rectangle is
of the parallelogram is the perpendicular distance between two parallel
Warning: Notice that the height
sides of the parallelogram, not a side of the parallelogram (unless the parallelogram is also a rectangle, of
course).
Area of a Parallelogram
If a parallelogram has base
units and height
units, then the area,
, is
square units.
Triangle
Example 5
How could we find the area of this triangle?
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Make it into a parallelogram. This can be done by making a copy of the original triangle and putting the
copy together with the original.
The area of the parallelogram is
, so the area of the triangle is
or
Warning: Notice that the height
(also often called the altitude) of the triangle is the perpendicular distance
between a vertex and the opposite side of the triangle.
Area of a Triangle
If a triangle has base
units and altitude
units, then the area,
, is
or
square units.
or
Lesson Summary
Once we understood the meaning of measures of space in two dimensions—in other words, area—we saw
the advantage of using square units. With square units established, the formula for the area of a rectangle
is simply a matter of common sense. From that point forward, the formula for the area of each new figure
builds on the previous figure. For a parallelogram, convert it to a rectangle. For a triangle, double it to make
a parallelogram.
Points to Consider
As we study other figures, we will frequently return to the basics of this lesson—the benefit of square units,
and the fundamental formula for the area of a rectangle.
It might be interesting to note that the word geometry is derived from ancient Greek roots that mean Earth
(geo-) measure (-metry). In ancient times geometry was very similar to today’s surveying of land. You can
see that land surveying became easily possible once knowledge of how to find the area of plane figures
was developed.
Lesson Exercises
Complete the chart. Base and height are given in units; area is in square units.
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Base
Height
Area
1.
?
2.
?
3.
?
4.
?
?
5.
6.
?
-foot by
-foot room cost
. The same kind of carpet cost
7. The carpet for a
with a square floor. What are the dimensions of the room?
for a room
8. Explain how an altitude of a triangle can be outside the triangle.
9. Line
and line
are parallel.
Explain how you know that
, and
all have the same area.
10. Lin bought a tract of land for a new apartment complex. The drawing below shows the measurements
of the sides of the tract. Approximately how many acres of land did Lin buy?
feet.)
acre
square
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11.
A
hexagon
is
drawn
on
a
coordinate
grid.
and
The
vertices
of
the
hexagon
are
What is the area of
Answers
1.
2.
3.
4.
5.
6.
7.
feet by
feet
8. This happens in a triangle with an obtuse angle. Each altitude to a side of the obtuse angle is outside the
triangle.
9. All of the triangles have the same base and altitude, so in each triangle
other triangles.
10.
sq ft
is the same as in each of the
acres
11.
Trapezoids, Rhombi, and Kites
Learning Objectives
•
Understand the relationships between the areas of two categories of quadrilaterals: basic quadrilaterals
(rectangles and parallelograms), and special quadrilaterals (trapezoids, rhombi, and kites).
•
Derive area formulas for trapezoids, rhombi, and kites.
•
Apply the area formulas for these special quadrilaterals.
Introduction
We’ll use the area formulas for basic shapes to work up to the formulas for special quadrilaterals. It’s an
easy job to convert a trapezoid to a parallelogram. It’s also easy to take apart a rhombus or kite and rebuild
622
it as a rectangle. Once we do this, we can derive new formulas from the old ones.
We’ll also need to review basic facts about the trapezoid, rhombus, and kite.
Area of a Trapezoid
Recall that a trapezoid is a quadrilateral with one pair of parallel sides. The lengths of the parallel sides are
the bases. The perpendicular distance between the parallel sides is the height, or altitude, of the trapezoid.
To find the area of the trapezoid, turn the problem into one about a parallelogram. Why? Because you already
know how to compute the area of a parallelogram.
Make a copy of the trapezoid.
Rotate the copy
.
Put the two trapezoids together to form a parallelogram.
Two things to notice:
1. The parallelogram has a base that is equal to
+
.
2. The altitude of the parallelogram is the same as the altitude of the trapezoid.
Now to find the area of the trapezoid:
The area of the parallelogram is base
altitude =
.
The parallelogram is made up of two congruent trapezoids, so the area of each trapezoid is one-half
the area of the parallelogram.
The area of the trapezoid is one-half of
.
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Area of Trapezoid with Bases
Trapezoid with bases
and
and
and Altitude
and altitude
or
Notice that the formula for the area of a trapezoid could also be written as the "Average of the bases time
the height." This may be a convenient shortcut for memorizing this formula.
Example 1
What is the area of the trapezoid below?
The bases of the trapezoid are
and
. The altitude is
.
Area of a Rhombus or Kite
First let’s start with a review of some of the properties of rhombi and kites.
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Kite
Rhombus
Congruent sides
Pairs
Opposite angles congruent
Pair yes.
Perpendicular diagonals
All
Pair maybe
Yes
Diagonals bisected
Yes.
Both pairs yes
Yes
maybe
Both yes
Now you’re ready to develop area formulas. We’ll follow the command: “Frame it in a rectangle.” Here’s how
you can frame a rhombus in a rectangle.
Notice that:
The base and height of the rectangle are the same as the lengths of the two diagonals of the
rhombus.
The rectangle is divided into congruent triangles;
the rhombus is one-half the area of the rectangle.
Area of a Rhombus with Diagonals
of the triangles fill the rhombus, so the area of
and
We can go right ahead with the kite. We’ll follow the same command again: “Frame it in a rectangle.” Here’s
how you can frame a kite in a rectangle.
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Notice that:
The base and height of the rectangle are the same as the lengths of the two diagonals of the kite.
The rectangle is divided into triangles; of the triangles fill the kite. For every triangle inside the
kite, there is a congruent triangle outside the kite so the area of the kite is one-half the area of the
rectangle.
Area of a Kite with Diagonals
and
Lesson Summary
We see the principle of “no need to reinvent the wheel” in developing the area formulas in this section. If we
wanted to find the area of a trapezoid, we saw how the formula for a parallelogram gave us what we needed.
In the same way, the formula for a rectangle was easy to modify to give us a formula for rhombi and kites.
One of the striking results is that the same formula works for both rhombi and kites.
Points to Consider
You’ll use area concepts and formulas later in this course, as well as in real life.
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•
Surface area of solid figures: the amount of outside surface.
•
Geometric probability: chances of throwing a dart and landing in a given part of a figure.
•
Carpet for floors, paint for walls, fertilizer for a lawn, and more: areas needed.
Tech Note - Geometry Software
You saw earlier that the area of a rhombus or kite depends on the lengths of the diagonals.
This means that all rhombi and kites with the same diagonal lengths have the same area.
Try using geometry software to experiment as follows.
•
Construct two perpendicular segments.
•
Adjust the segments so that one or both of the segments are bisected.
•
Draw a quadrilateral that the segments are the diagonals of. In other words, draw a quadrilateral for
which the endpoints of the segments are the vertices.
•
Repeat with the same perpendicular, bisected segments, but making a different rhombus or kite. Repeat
for several different rhombi and kites.
•
Regardless of the specific shape of the rhombus or kite, the areas are all the same.
The same activity can be done on a geoboard. Place two perpendicular rubber bands so that one or both
are bisected. Then place another rubber band to form a quadrilateral with its vertices at the endpoints of the
two segments. A number of different rhombi and kites can be made with the same fixed diagonals, and
therefore the same area.
Lesson Exercises
Quadrilateral
has vertices
1. Show that
is a trapezoid.
2. What is the area of
and
in a coordinate plane.
?
3. Prove that the area of a trapezoid is equal to the area of a rectangle with height the same as the height
of the trapezoid and base equal to the length of the median of the trapezoid.
4. Show that the trapezoid formula can be used to find the area of a parallelogram.
5. Sasha drew this plan for a wood inlay he is making.
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is the length of the slanted side.
is a rhombus.
is the length of the horizontal line segment. Each shaded section
The shaded sections are rhombi. Based on the drawing, what is the total area of the shaded sections?
6. Plot
points on a coordinate plane.
•
The points are the vertices of a rhombus.
•
The area of the rhombus is
square units.
7. Tyra designed the logo for a new company. She used three congruent kites.
What is the area of the entire logo?
8. In the figure below:
is a square
feet
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What is the area of
?
In the figure below:
is a square
feet
feet
9. What is the area of
10. The area of
?
is what fractional part of the area of
?
Answers
1. Slope of
, slope of
629
are parallel.
2.
slope of
and
are the bases,
4. For a parallelogram,
the area is
=
is an altitude.
(the “bases” are two of the parallel sides), so by the trapezoid formula
5. Length of long diagonal of one rhombus is
of
. Length of other diagonal is
(each rhombus is made
right triangles).
Total area is
6. Many rhombi work, as long as the product of the lengths of the diagonals is
.
7.
8.
square feet
9.
square feet
10.
Areas of Similar Polygons
Learning Objectives
630
•
Understand the relationship between the scale factor of similar polygons and their areas.
•
Apply scale factors to solve problems about areas of similar polygons.
•
Use scale models or scale drawings.
Introduction
We’ll begin with a quick review of some important features of similar polygons. You remember that we
studied similar figures rather extensively in Chapter 7. There you learned about scale factors and perimeters
of similar polygons. In this section we’ll take similar figures one step farther. We’ll see that the areas of
similar figures have a very specific relationship to the scale factor—but it’s just a bit tricky! We wrap up the
section with some thoughts on why living things are the “right” size, and what geometry has to do with that!
Review - Scale Factors and Perimeter
Example 1
The diagram below shows two rhombi.
a. Are the rhombi similar? How do you know?
Yes.
•
The sides are parallel, so the corresponding angles are congruent.
•
Using the Pythagorean Theorem, we can see that each side of the smaller rhombus has a length of
, and each side of the larger rhombus has a length of
.
•
So the lengths of the sides are proportional.
•
Polygons with congruent corresponding angles and proportional sides are similar.
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b. What is the scale factor relating the rhombi?
The scale factor relating the smaller rhombus to the larger one is
c. What is the perimeter of each rhombus?
Answer
•
Perimeter of smaller rhombus
•
Perimeter of larger rhombus
d. What is the ratio of the perimeters?
e. What is the area of each rhombus?
Area of smaller rhombus =
Area of larger rhombus =
What do you notice in this example? The perimeters have the same ratio as the scale factor.
But what about the areas? The ratio of the areas is certainly not the same as the scale factor. If it were, the
area of the larger rhombus would be
, but the area of the larger rhombus is actually
What IS the ratio of the areas?
The ratio of the areas is
Notice that
or in decimal,
.
So at least in this case we see that the ratio of the areas is the square of the scale factor.
Scale Factors and Areas
What happened in Example 1 is no accident. In fact, this is the basic relationship for the areas of similar
polygons.
Areas of Similar Polygons
If the scale factor relating the sides of two similar polygons is
the larger polygon is
times the area of the smaller polygon. In symbols let the area
of the smaller polygon be
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, then the area of
and the area of the larger polygon be
. Then:
Think about the area of a polygon. Imagine that you look at a square with an area of exactly square unit.
Of course, the sides of the square are unit of length long. Now think about another polygon that is similar
to the first one with a scale factor of
. Every -by- square in the first polygon has a matching
-bysquare in the second polygon, and the area of each of these
reasoning, every
-by-
squares is
square unit of area in the first polygon has a corresponding
polygon. So the total area of the second polygon is
. Extending this
units of area in the second
times the area of the first polygon.
Warning: In solving problems it’s easy to forget that you do not always use just the scale factor. Use the
scale factor in problems about lengths. But use the square of the scale factor in problems about area!
Example 2
Wu and Tomi are painting murals on rectangular walls. The length and width of Tomi’s wall are
length and width of Wu’s wall.
a. The total length of the border of Tomi’s wall is
wall?
times the
feet. What is the total length of the border of Wu’s
This is a question about lengths, so you use the scale factor itself. All the sides of Tomi’s wall are times
the length of the corresponding side of Wu’s wall, so the perimeter of Tomi’s wall is also
times the
perimeter of Wu’s wall.
The total length of the border (perimeter) of Wu’s wall is
b. Wu can cover his wall with
feet.
quarts of paint. How many quarts of paint will Tomi need to cover her wall?
This question is about area, since the area determines the amount of paint needed to cover the walls. The
ratio of the amounts of paint is the same as the ratio of the areas (which is the square of the scale factor).
Let
be the amount of paint that Tomi needs.
Tomi would need
quarts of paint.
Summary of Length and Area Relationships for Similar Polygons
If two similar polygons are related by a scale factor of
•
, then:
Length: The lengths of any corresponding parts have the same ratio,
*Area: The ratio of the areas is
. Note that this applies to sides,
. Note that this applies to areas, and any aspect of an object that
Note: You might be able to make a pretty good guess about the volumes of similar solid
You’ll see more about that in Chapter 11.
figures.
Scale Drawings and Scale Models
One important application of similar figures is the use of scale drawings and scale models. These are twodimensional (scale drawings) or three-dimensional (scale models) representations of real objects. The
drawing or model is similar to the actual object.
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Scale drawings and models are widely used in design, construction, manufacturing, and many other fields.
Sometimes a scale is shown, such as “ inch
miles” on a map. Other times the scale may be calculated,
if necessary, from information about the object being modeled.
Example 3
Jake has a map for a bike tour. The scale is
tour were about
inch
miles. He estimated that two scenic places on the
inches apart on the map. How far apart are these places in reality?
Each inch on the map represents a distance of
miles. The places are about
miles apart.
Example 4
Cristy’s design team built a model of a spacecraft to be built. Their model has a scale of
spacecraft will be
feet long. How long should the model be?
Let
. The actual
be the length of the model.
The model should be
feet long.
Example 5
Tasha is making models of several buildings for her senior project. The models are all made with the same
scale. She has started the chart below.
a. What is the scale of the models?
The scale is
inch
feet.
b. Complete the chart below.
Building
Actual height (feet) Model height (inches)
Sears Tower
?
(Chicago)
Empire State Building
(New York City)
Columbia Center
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?
(Seattle)
. It is
Sears Tower:
Columbia Center: Let
feet high.
= the model height.
The model should be about
inches high.
Why There Are No 12-Foot-Tall Giants
Why are there no
-foot-tall giants? One explanation for this is a matter of similar figures.
Let’s suppose that there is a
-foot-tall human. Compare this giant (?) to a
apply some facts about similar figures.
The scale factor relating these two hypothetical people is
scale factor.
All linear dimensions of the giant would be
This includes height, bone length, etc.
-foot-tall person. Now let’s
. Here are some consequences of this
times the corresponding dimensions of the real person.
All area measures of the giant would be
times the corresponding area measures of the real
person. This includes respiration (breathing) and metabolism (converting nutrients to usable materials
and energy) rates, because these processes take place along surfaces in the lungs, intestines, etc.
This also includes the strength of bones, which depends on the cross-section area of the bone.
All volume measures of the giant would be
times the corresponding volume measures of
the real person. (You’ll learn why in Chapter 11.) The volume of an organism generally determines
its weight and mass.
What kinds of problems do we see for our giant? Here are two severe ones.
1. The giant would have bones that are times as strong, but those bones have to carry a body weight that
is times as much. The bones would not be up to the task. In fact it appears that the giant’s own weight
would be able to break its bones.
2. The giant would have times the weight, number of cells, etc. of the real person, but only
much ability to supply the oxygen, nutrition, and energy needed.
Conclusion: There are no
ometry of similar figures.
times as
-foot-giants, and some of the reasons are nothing more, or less, than the ge-
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For further reading: On Being the Right Size, by J. B. S. Haldane, also available at http://irl.cs.ucla.edu/papers/right-size.html.
Lesson Summary
In his lesson we focused on one main point: The areas of similar polygons have a ratio that is the square
of the scale factor. We also used ideas about similar figures to analyze scale drawings and scale models,
which are actually similar representations of actual objects.
Points to Consider
You have now learned quite a bit about the lengths of sides and areas of polygons. Next we’ll build on
knowledge about polygons to come to a conclusion about the “perimeter” of the “ultimate polygon,” which
is the circle.
Suppose we constructed regular polygons that are all inscribed in the same circle.
•
Think about polygons that have more and more sides.
•
How would the perimeter of the polygons change as the number of sides increases?
The answers to these questions will lead us to an understanding of the formula for the circumference
(perimeter) of a circle.
Lesson Exercises
The figure below is made from small congruent equilateral triangles.
congruent small triangles fit together to make a bigger, similar triangle.
1. What is the scale factor of the large and small triangles?
2. If the area of the large triangle is
square units, what is the area of a small triangle?
The smallest squares in the diagram below are congruent.
3. What is the scale factor of the shaded square and the largest square?
4. If the area of the shaded square is
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square units, what is the area of he largest square?
5. Frank drew two equilateral triangles. Each side of one triangle is
times as long as a side of the other
triangle. The perimeter of the smaller triangle is
cm. What is the perimeter of the larger triangle?
In the diagram below, .
.
6. What is the scale factor of the small triangle and the large triangle?
, what is the perimeter of the small triangle?
7. If the perimeter of the large triangle is
8. If the area of the small triangle is
, write an expression for the area of the large triangle.
9. If the area of the small triangle is
, write an expression for the area of the trapezoid.
10. The area of one square on a game board is exactly twice the area of another square. Each side of the
mm long. How long is each side of the smaller square?
larger square is
11. The distance from Charleston to Morgantown is
miles. The distance from Fairmont to Elkins is
miles. Charleston and Morgantown are inches apart on a map. How far apart are Fairmont and Elkins on
the same map?
Marlee is making models of historic locomotives (train engines). She uses the same scale for all of her
models.
The
The
locomotive was
feet long. The model is
Class locomotive was
inches long.
feet long.
12. What is the scale of Marlee’s models?
13. How long is the model of the
Class locomotive?
Answers
1.
2.
3.
or
4.
637
5.
cm
6.
7.
8.
or
9.
or
10.
mm
11.
inches
12.
13.
inch
feet or equivalent
inches
Circumference and Arc Length
Learning Objectives
•
Understand the basic idea of a limit.
•
Calculate the circumference of a circle.
•
Calculate the length of an arc of a circle.
Introduction
In this lesson, we extend our knowledge of perimeter to the perimeter—or circumference—of a circle. We’ll
use the idea of a limit to derive a well-known formula for the circumference. We’ll also use common sense
to calculate the length of part of a circle, known as an arc.
The Parts of a Circle
A circle is the set of all points in a plane that are a given distance from another point called the center. Flat
round things, like a bicycle tire, a plate, or a coin, remind us of a circle.
638
The diagram reviews the names for the “parts” of a circle.
•
The center
•
The circle: the points that are a given distance from the center (which does not include the center or interior)
•
The interior: all the points (including the center) that are inside the circle
•
circumference: the distance around a circle (exactly the same as perimeter)
•
radius: any segment from the center to a point on the circle (sometimes “radius” is used to mean the
length of the segment and it is usually written as )
•
diameter: any segment from a point on the circle, through the center, to another point on the circle
(sometimes “diameter” is used to mean the length of the segment and it is usually written as
)
If you like formulas, you can already write one for a circle:
or
Circumference Formula
The formula for the circumference of a circle is a classic. It has been known, in rough form, for thousands
of years. Let’s look at one way to derive this formula.
Start with a circle with a diameter of unit. Inscribe a regular polygon in the circle. We’ll inscribe regular
polygons with more and more sides and see what happens. For each inscribed regular polygon, the
perimeter will be given (how to figure that is in a review question).
What do you notice?
1. The more sides there are, the closer the polygon is to the circle itself.
2. The perimeter of the inscribed polygon increases as the number of sides increases.
3. The more sides there are, the closer the perimeter of the polygon is to the circumference of the circle.
Now imagine that we continued inscribing polygons with more and more sides. It would become nearly impossible to tell the polygon from the circle. The table below shows the results if we did this.
Regular Polygons Inscribed in a Circle with Diameter
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Number of sides of polygon Perimeter of polygon
3.141
As the number of sides of the inscribed regular polygon increases, the perimeter seems to approach a “limit.”
This limit, which is the circumference of the circle, is approximately
. This is the famous and well.
is an endlessly non-repeating decimal number. We often use
as a value
known number
for
in calculations, but this is only an approximation.
Conclusion: The circumference of a circle with diameter
is
.
For Further Reading
Mathematicians have calculated the value of
to thousands, and even millions, of decimal places. You
might enjoy finding some of these megadecimal numbers. Of course, all are approximately equal to
.
The article at the following URL shows more than a million digits of the decimal for
.
http://wiki.answers.com/Q/What_is_the_exact_value_for_Pi_at_this_moment
Tech Note - Geometry Software
You can use geometry software to continue making more regular polygons inscribed
in a circle with diameter and finding their perimeters.
Can we extend this idea to other circles? First, recall that all circles are similar to each other. (This is also
true for all equilateral triangles, all squares, all regular pentagons, etc.)
Suppose a circle has a diameter of
•
The scale factor of this circle and the one in the diagram and table above, with diameter
, or just
•
units.
,
.
You know how a scale factor affects linear measures, which include perimeter and circumference. If the
scale factor is
, then the perimeter is
times as much.
This means that if the circumference of a circle with diameter
with diameter
is
.
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, is
is
, then the circumference of a circle
Circumference Formula
Let
be the diameter of a circle, and
the circumference.
Example 1
A circle is inscribed in a square. Each side of the square is
circle?
Use
cm long. What is the circumference of the
. The length of a side of the square is also the diameter of the circle.
Note that sometimes an approximation is given using
. In this example the circumference is
cm using that approximation. An exact is given in terms of (leaving the symbol for in the answer rather
than multiplying it out. In this example the exact circumference is
.
Arc Length
Arcs are measured in two different ways.
•
Degree measure: The degree measure of an arc is the fractional part of a
the arc is.
•
Linear measure: This is the length, in units such as centimeters and feet, if you traveled from one end
of the arc to the other end.
complete circle that
Example 2
Find the length of
.
=
. The radius of the circle is
inches.
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Remember,
is
is the measure of the central angle associated with
.
of a circle. The circumference of the circle is
inches. The arc length of
is
inches.
In this lesson we study the second type of arc measure—the measure of an arc’s length. Arc length is directly
related to the degree measure of an arc.
Suppose a circle has:
•
circumference
•
diameter
•
radius
Also, suppose an arc of the circle has degree measure
Note that
.
is the fractional part of the circle that the arc represents.
Arc length
Lesson Summary
This lesson can be summarized with a list of the formulas developed.
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•
Radius and diameter:
•
Circumference of a circle:
•
Arc length =
Points to Consider
After perimeter and circumference, the next logical measure to study is area. In this lesson, we learned
about the perimeter of a circle (circumference) and the arc length of a sector. In the next lesson we’ll learn
about the areas of circles and sectors.
Lesson Exercises
1. Prove: The circumference of a circle with radius
is
2. The Olympics symbol is five congruent circles arranged as shown below. Assume the top three circles
are tangent to each other.
Brad is tracing the entire symbol for a poster. How far will his pen point travel?
3. A truck has tires that measure
inches from the center of the wheel to the outer edge of the tire.
a. How far forward does the truck travel every time a tire turns exactly once?
b. How many times will the tire turn when the truck travels
mile?
4. The following wire sculpture was made from two perpendicular
at the center of a circle.
mile
feet).
cm segments that intersect each other
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a. If the radius of the circle is
cm, how much wire was used to outline the shaded sections?
5. The circumference of a circle is
feet. What is the radius of the circle?
6. A gear with a radius of inches turns at a rate of
point on the edge of the pulley travel in one second?
RPM (revolutions per minute). How far does a
m long. The boom is anchored at the center pivot.
7. A center pivot irrigation system has a boom that is
It revolves around the center pivot point once every three days. How far does the tip of the boom travel in
one day?
8. The radius of Earth at the Equator is about
miles. Belem (in Brazil) and the Galapagos Islands (in
the Pacific Ocean) are on (or very near) the Equator. The approximate longitudes are Belem,
, and
Galapagos Islands,
.
a. What is the degree measure of the major arc on the Equator from Belem to the Galapagos Islands?
b. What is the distance from Belem to the Galapagos Islands on the Equator the “long way around?”
sides. Write a formula that expresses the
9. A regular polygon inscribed in a circle with diameter has
perimeter, , of the polygon in terms of
. (Hint: Use trigonometry.)
10. The pulley shown below revolves at a rate of
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RPM.
a. How far does point
travel in one hour?
Answers
1.
2.
inches
3.
a.
inches
b. Approximately
4.
times
cm
5. Approximately
feet
6. Approximately
inches
7. Approximately
8.
a.
b. Approximately
miles
or equivalent
9.
10.
Circles and Sectors
Learning Objectives
•
Calculate the area of a circle.
•
Calculate the area of a sector.
•
Expand understanding of the limit concept.
Introduction
In this lesson we complete our area toolbox with formulas for the areas of circles and sectors. We’ll start
with areas of regular polygons, and work our way to the limit, which is the area of a circle. This may sound
familiar; it’s exactly the same approach we used to develop the formula for the circumference of a circle.
Area of a Circle
The big idea:
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•
Find the areas of regular polygons with radius
•
Let the polygons have more and more sides.
•
See if a limit shows up in the data.
•
Use similarity to generalize the results.
.
The details:
Begin with polygons having
and
sides, inscribed in a circle with a radius of
.
Now imagine that we continued inscribing polygons with more and more sides. It would become nearly impossible to tell the polygon from the circle. The table below shows the results if we did this.
Regular Polygons Inscribed in a Circle with Radius
Number of sides of polygon Area of polygon
As the number of sides of the inscribed regular polygon increases, the area seems to approach a “limit.”
This limit is approximately
, which is
.
Conclusion: The area of a circle with radius
is
.
Now we extend this idea to other circles. You know that all circles are similar to each other.
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Suppose a circle has a radius of
units.
•
The scale factor of this circle and the one in the diagram and table above, with radius
or just .
•
You know how a scale factor affects area measures. If the scale factor is
as much.
This means that if the area of a circle with radius
is
, is
,
, then the area is
, then the area of a circle with radius
times
is
.
Area of a Circle Formula
Let
be the radius of a circle, and
the area.
You probably noticed that the reasoning about area here is very similar to the reasoning in an earlier lesson
when we explored the perimeter of polygons and the circumference of circles.
Example 1
A circle is inscribed in a square. Each side of the square is
Use
cm long. What is the area of the circle?
. The length of a side of the square is also the diameter of the circle. The radius is
The area is
.
.
Area of a Sector
The area of a sector is simply an appropriate fractional part of the area of the circle. Suppose a sector of a
circle with radius
and circumference
has an arc with a degree measure of
and an arc length of
units.
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•
The sector is
of the circle.
•
The sector is also
of the circle.
To find the area of the sector, just find one of these fractional parts of the area of the circle. We know that
the area of the circle is
. Let
be the area of the sector.
Also,
Area of a Sector
. A sector of the circle has an arc with degree measure
A circle has radius
and arc length units.
The area of the sector is
square units.
Example 2
Mark drew a sheet metal pattern made up of a circle with a sector cut out. The pattern is made from an arc
of a circle and two perpendicular -inch radii.
How much sheet metal does Mark need for the pattern?
The measure of the arc of the piece is
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, which is
of the circle.
The area of the sector (pattern) is =
sq in.
Lesson Summary
We used the idea of a limit again in this lesson. That enabled us to find the area of a circle by studying
polygons with more and more sides. Our approach was very similar to the one used earlier for the circumference of a circle. Once the area formula was developed, the area of a sector was a simple matter of taking
the proper fractional part of the whole circle.
Summary of Formulas:
Area Formula
Let be the radius of a circle, and
Area of a Sector
A circle has radius
the area.
. A sector of the circle has an arc with degree measure
and arc length
units.
square units.
The area of the sector is
Points to Consider
When we talk about a limit, for example finding the limit of the areas of regular polygons, how many sides
do we mean when we talk about “more and more?” As the polygons have more and more sides, what happens
to the length of each side? Is a circle a polygon with an infinite number of sides? And is each “side” of a
circle infinitely small? Now that’s small!
In the next lesson you’ll see where the formula comes from that gives us the areas of regular polygons. This
is the formula that was used to produce the table of areas in this lesson.
Lesson Exercises
Complete the table of radii and areas of circles. Express your answers in terms of
Radius
Area
(units)
(square units)
1.
?
2.
?
3.
?
4.
.
?
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5. Prove: The area of a circle with diameter
is
6. A circle is inscribed in a square.
The yellow shaded area is what percent of the square?
7. The circumference of a circle is
feet. What is the area of the circle?
8. A center pivot irrigation system has a boom that is
long. The boom is anchored at the center pivot.
It revolves around the center pivot point once every three days, irrigating the ground as it turns. How many
hectares of land are irrigated each day?
(
hectare
)
9. Vicki is cutting out a gasket in her machine shop. She made a large circle of gasket material, then cut out
and removed the two small circles. The centers of the small circles are on a diameter of the large circle.
Each square of the grid is 1 square inch.
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How much gasket material will she use for the gasket?
10. A security system scans all points up to
of
.
from is base. It scans back and forth through an angle
How much space does the system cover?
11. A simplified version of the international radiation symbol is shown below.
(Source: http://upload.wikimedia.org/wikipedia/commons/0/0b/Radiation_warning_symbol.svg
Public Domain)
License:
The symbol is made from two circles and three equally spaced diameters of the large circle. The diameter
of the large circle is
inches, and the diameter of the small circle is inches. What is the total area of the
symbol?
12. Chad has
feet of fencing. He will use it all. Which would enclose the most space, a square fence
or a circular fence? Explain your answer.
Answers
1.
2.
3.
4.
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5.
6. Approximately
7. Approximately
square feet
8. Approximately
9. Approximately
square inches
10. Approximately
11.
square inches
12. The circular fence has a greater area.
Square:
Circle:
Regular Polygons
Learning Objectives
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•
Recognize and use the terms involved in developing formulas for regular polygons.
•
Calculate the area and perimeter of a regular polygon.
•
Relate area and perimeter formulas for regular polygons to the limit process in prior lessons.
Introduction
You’ve probably been asking yourself, “Where did the areas and perimeters of regular polygons in earlier
lessons come from?” Or maybe not! You might be confident that the information presented then was accurate.
In either case, in this lesson we’ll fill in the missing link. We’ll derive formulas for the perimeter and area of
any regular polygon.
You already know how to find areas and perimeters of some figures—triangles, rectangles, etc. Not surprisingly, the new formulas in this lesson will build on those basic figures—in particular, the triangle. Note too
that we will find an outstanding application of trigonometric functions in this lesson.
Parts and Terms for Regular Polygons
Let’s start with some background on regular polygons.
Here is a general regular polygon with
sides; some of its sides are shown.
In the diagram, here is what each variable represents.
is the length of each side of the polygon.
is the length of a “radius” of the polygon, which is a segment from a vertex of the polygon to the
center.
is the length of one-half of a side of the polygon
is the length of a segment called the apothem—a segment from the center to a side of the polygon,
perpendicular to the side. (Notice that
is the altitude of each of the triangles formed by two radii
and a side.)
The angle between two consecutive radii measures
because
congruent central angles are formed
by the radii from the center to each of the
vertices of the polygon. An apothem divides each of these
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central angles into two congruent halves; each of these half angles measures
.
Using Trigonometry with the Regular Polygon
Recall that in a right triangle:
sine of an angle =
cosine of an angle =
In the diagram above, for the half angles mentioned,
is the length of the opposite side
is the length of the adjacent side
is the length of the hypotenuse
Now we can put these facts together:
•
•
•
•
Perimeter of a Regular Polygon
We continue with the regular polygon diagrammed above. Let
be the perimeter. In simplest terms,
Here is an alternate version of the perimeter formula.
Perimeter of a regular polygon with
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sides and a radius
units long:
One more version of the perimeter formula applies when the polygon is inscribed in a “unit circle,” which is
a circle with a radius of .
Perimeter of a regular polygon with
sides inscribed in a unit circle:
Example 1
A square has a radius of
Use
inches. What is the perimeter of the square?
, with
and
.
Notice that a side and two radii make a right triangle.
The legs are
inches long, and the hypotenuse, which is a side of the square, is
inches long.
Use
inches.
The purpose of this example is not to calculate the perimeter, but to verify that the formulas developed above
“work.”
Area of a Regular Polygon
The next logical step is to complete our study of regular polygons by developing area formulas.
Take another look at the regular polygon figure above. Here’s how we can find its area,
Two radii and a side make a triangle with base
•
There are
and altitude
.
.
of these triangles.
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•
The area of each triangle is
.
The entire area is
.
Area of a regular polygon with apothem
:
We can use trigonometric functions to produce a different version of the area formula.
(remember that
(remember that
Area of a regular polygon with
sides and radius
)
and
:
One more version of the area formula applies when the polygon is inscribed in a unit circle.
(remember that
)
Area of a regular polygon with
sides inscribed in a unit circle:
Example 2
A square is inscribed in a unit circle. What is the area of the square?
Use
with
The square is a rhombus with diagonals
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.
units long. Use the area formula for a rhombus.
)
Comments: As in example 1, the purpose of this example is to show that the new area formulas do work.
We can confirm that the area formula gives a correct answer because we have another way to confirm that
the area is correct.
Lesson Summary
The lesson can be summarized with a review of the formulas we derived.
Perimeter
Area
Any regular polygon
Any regular polygon
Regular polygon inscribed in a unit circle
Points to Consider
We used the concept of a limit in an earlier lesson. In the Lesson Exercises, you will have an opportunity to
use the formulas from this lesson to “confirm” the circumference and area formulas for a circle, which is the
“ultimate” regular polygon (with many, many sides that are very short).
Lesson Exercises
Each side of a regular hexagon is
inches long.
1. What is the radius of the hexagon?
2. What is the perimeter of the hexagon?
3. What is the area of the hexagon?
A regular
-gon and a regular
-gon are inscribed in a circle with a radius of
centimeters.
4. Which polygon has the greater perimeter?
5. How much greater is the perimeter?
6. Which polygon has the greater area?
7. How much greater is the area?
-gon is inscribed in a unit circle. The area of the
8. A regular
is
. What is the smallest possible value of
?
-gon, rounded to the nearest hundredth,
Answers
1.
2.
inches
inches
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3.
square inches
4. The
5.
-gon
cm
6. The
-gon
7.
8.
Geometric Probability
Learning Objectives
•
Identify favorable outcomes and total outcomes.
•
Express geometric situations in probability terms.
•
Interpret probabilities in terms of lengths and areas.
Introduction
You’ve probably studied probability before now (pun intended). We’ll start this lesson by reviewing the basic
concepts of probability.
Once we’ve reviewed the basic ideas of probability, we’ll extend them to situations that are represented in
geometric settings. We focus on probabilities that can be calculated based on lengths and areas. The formulas you learned in earlier lessons will be very useful in figuring these geometric probabilities.
Basic Probability
Probability is a way to assign specific numbers to how likely, or unlikely, an event is. We need to know two
things:
•
•
the number of “favorable” outcomes for the event. Let’s call this
The probability of the event, call it
of outcomes.
Definition of Probability
Example 1
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.
the total number of possible outcomes for an event. Let’s call this
.
, is the ratio of the number of favorable outcomes to the total number
Nabeel’s company has
holidays each year. Holidays are always on weekdays (not weekends). This
year there are
weekdays. What is the probability that any weekday is a holiday?
There are
weekdays in all.
of the weekdays are holidays
Comments: Probabilities are often expressed as fractions, decimals, and percents. Nabeel can say that
there is a
probability.
chance of any weekday being a holiday. Note that this is (unfortunately?) a relatively low
Example 2
Charmane has four coins in a jar: two nickels, a dime, and a quarter. She mixes them well. Charmane takes
out two of the coins without looking. What is the probability that the coins she takes have a total value of
more than
?
in this problem is the total number of two-coin combinations. We can just list them all. To make it easy
to keep track, use these codes:
quarter).
(one of the nickels),
(the other nickel),
(the dime), and
(the
Two-coin combinations:
There are six two-coin combinations.
Of the six two-coin combinations, three have a total value of more than
The
probability
that
the
two
coins
will
have
a
total
value
. They are:
of
more
than
is
.
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The probability is usually written as
, or
. Sometimes this is expressed as “a 50-50 chance”
because the probability of success and of failure are both
.
Geometric Probability
The values of
and
that determine a probability can be lengths and areas.
Example 3
Sean needs to drill a hole in a wall that is
feet wide and feet high. There is a 2-foot-by-3-foot rectangular mirror on the other side of the wall so that Sean can’t see the mirror. If Sean drills at a random location
on the wall, what is the probability that he will hit the mirror?
The area of the wall is
The area of the mirror is
square feet. This is
square feet. This is
.
.
.
The probability is
Example 4
Ella repairs an electric power line that runs from Acton to Dayton through Barton and Canton. The distances
in miles between these towns are as follows.
•
Barton to Canton = 8 miles.
•
Acton to Canton = 12 miles.
•
Canton to Dayton = 2 miles.
If a break in the power line happens, what is the probability that the break is between Barton and Dayton?
Approximately
.
= the distance from Acton to Dayton
= the distance from Barton to Dayton
miles.
miles.
Lesson Summary
Probability is a way to measure how likely or unlikely an event is. In this section we saw how to use lengths
and areas as models for probability questions. The basic probability ideas are the same as in non-geometry
applications, with probability defined as:
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Points to Consider
Some events are more likely, and some are less likely. No event has a negative probability! Can you think
of an event with an extremely low, or an extremely high, probability? What are the ultimate extremes—the
greatest and the least values possible for a probability? In ordinary language these are called “impossible”
(least possible probability) and “certain” or a “sure thing” (greatest possible probability).
The study of probability originated in the seventeenth century as mathematicians analyzed games of chance.
For Further Reading
French mathematicians Pierre de Fermat and Blaise Pascal are credited as the “inventors” of mathematical
probability. The reference below is an easy introduction to their ideas.
http://mathforum.org/isaac/problems/prob1.html
Lesson Exercises
1. Rita is retired. For her, every day is a holiday. What is the probability that tomorrow is a holiday for Rita?
2. Chaz is “on call” any time, any day. He never has a holiday. What is the probability that tomorrow is a
holiday for Chaz?
3. The only things on Ray’s refrigerator door are
magnet off without looking.
green magnets and
yellow magnets. Ray takes one
a. What is the probability that the magnet is green?
b. What is the probability that the magnet is yellow?
c. What is the probability that the magnet is purple?
Ray takes off two magnets without looking.
d. What is the probability that both magnets are green?
e. What is the probability that Ray takes off one green and one yellow magnet?
4. Reed uses the diagram below as a model of a highway.
Reed got a call about an accident at an unknown location between Acton and Dayton.
a. What is the probability that the accident is not between Canton and Dayton?
b. What is the probability that the accident is closer to Canton than it is to Barton?
5. A tire has an outer diameter of
inches. Nina noticed a weak spot on the tire. She marked the weak
spot with chalk. The chalk mark is inches along the outer edge of the tire. What is the probability that part
of the weak spot is in contact with the ground at any time?
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6. Mike set up a rectangular landing zone that measures
feet by
feet. He marked a circular helicopter
pad that measured
feet across at its widest in the landing zone. As a test, Mike dropped a package that
landed in the landing zone. What is the probability that the package landed outside the helicopter pad?
7. Fareed made a target for a game. The target is a
-foot-by-
-foot square. To win a player must hit a
smaller square in the center of the target. If the probability that players who hit the target win is
is the length of a side of the smaller square?
, what
8. Amazonia set off on a quest. She followed the paths shown by the arrows in the map.
Every time a path splits, Amazonia takes a new path at random. What is the probability that she ends up in
the cave?
Answers
1.
, or equivalent
2.
3.
a.
, or equivalent
b.
,or equivalent
c.
d.
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or equivalent
e.
or equivalent
4.
a.
b.
5. Approximately
6. Approximately
7. Approximately
8.
feet
or equivalent
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11. Surface Area and Volume
The Polyhedron
Learning Objectives
•
Identify polyhedra.
•
Understand the properties of polyhedra.
•
Use Euler’s formula solve problems.
•
Identify regular (Platonic) polyhedra.
Introduction
In earlier chapters you learned that a polygon is a two-dimensional (planar) figure that is made of three or
more points joined together by line segments. Examples of polygons include triangles, quadrilaterals, pentagons, or octagons. In general, an n-gon is a polygon with n sides. So a triangle is a 3-gon, or 3-sided
polygon, a pentagon is a 5-gon, or 5-sided polygon.
You can use polygons to construct a 3-dimensional figure called a polyhedron (plural: polyhedra). A
polyhedron is a 3-dimensional figure that is made up of polygon faces. A cube is an example of a polyhedron
and its faces are squares (quadrilaterals).
Polyhedron or Not
A polyhedron has the following properties:
•
It is a 3-dimensional figure.
•
It is made of polygons and only polygons. Each polygon is called a face of the polyhedron.
•
Polygon faces join together along segments called edges.
•
Each edge joins exactly two faces.
•
Edges meet in points called vertices.
•
There are no gaps between edges or vertices.
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Example 1
Is the figure a polyhedron?
Yes. A figure is a polyhedron if it has all of the properties of a polyhedron. This figure:
•
Is 3-dimensional.
•
Is constructed entirely of flat polygons (triangles and rectangles).
•
Has faces that meet in edges and edges that meet in vertices.
•
Has no gaps between edges.
•
Has no non-polygon faces (e.g., curves).
•
Has no concave faces.
Since the figure has all of the properties of a polyhedron, it is a polyhedron.
Example 2
Is the figure a polyhedron?
No. This figure has faces, edges, and vertices, but all of its surfaces are not flat polygons. Look at the end
surface marked A. It is flat, but it has a curved edge so it is not a polygon. Surface B is not flat (planar).
Example 3
Is the figure a polyhedron?
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No. The figure is made up of polygons and it has faces, edges, and vertices. But the faces do not fit together—the figure has gaps. The figure also has an overlap that creates a concave surface. For these reasons,
the figure is not a polyhedron.
Face, Vertex, Edge, Base
As indicated above, a polyhedron joins faces together along edges, and edges together at vertices. The
following statements are true of any polyhedron:
•
Each edge joins exactly two faces.
•
Each edge joins exactly two vertices.
To see why this is true, take a look at this prism. Each of its edges joins two faces along a single line segment.
Each of its edges includes exactly two vertices.
Let’s count the number of faces, edges, and vertices in a few typical polyhedra. The square pyramid gets
its name from its base, which is a square. It has 5 faces, 8 edges, and 5 vertices.
Other figures have a different number of faces, edges, and vertices.
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If we make a table that summarizes the data from each of the figures we get:
Figure
Vertices
Faces
Edges
Square pyramid
5
5
8
Rectangular prism
8
6
12
Octahedron
6
8
12
Pentagonal prism
10
7
15
Do you see a pattern? Calculate the sum of the number of vertices and edges. Then compare that sum to
the number of edges:
Figure
V
F
E
V+F
square pyramid
5
5
8
10
rectangular prism
8
6
12
14
octahedron
6
8
12
14
pentagonal prism
10
7
15
17
Do you see the pattern? The formula that summarizes this relationship is named after mathematician
Leonhard Euler. Euler’s formula says, for any polyhedron:
Euler's Formula for Polyhedra
or
You can use Euler’s formula to find the number of edges, faces, or vertices in a polyhedron.
Example 4
Count the number of faces, edges, and vertices in the figure. Does it conform to Euler’s formula?
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There are 6 faces, 12 edges, and 8 vertices. Using the formula:
So the figure conforms to Euler’s formula.
Example 5
In a 6-faced polyhedron, there are 10 edges. How many vertices does the polyhedron have?
Use Euler's formula.
Euler’s formula
Substitute values for f and e
Solve
There are 6 vertices in the figure.
Example 6
A 3-dimensional figure has 10 vertices, 5 faces, and 12 edges. It is a polyhedron? How do you know?
Use Euler's formula.
Euler’s formula
Substitute values for v, f, and e
Evaluate
The equation does not hold so Euler’s formula does not apply to this figure. Since all polyhedra conform to
Euler’s formula, this figure must not be a polyhedron.
Regular Polyhedra
Polyhedra can be named and classified in a number of ways—by side, by angle, by base, by number of
faces, and so on. Perhaps the most important classification is whether or not a polyhedron is regular or not.
You will recall that a regular polygon is a polygon whose sides and angles are all congruent.
A polyhedron is regular if it has the following characteristics:
•
All faces are the same.
•
All faces are congruent regular polygons.
•
The same number of faces meet at every vertex.
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•
The figure has no gaps or holes.
•
The figure is convex—it has no indentations.
Example 7
Is a cube a regular polyhedron?
All faces of a cube are regular polygons—squares. The cube is convex because it has no indented surfaces.
The cube is simple because it has no gaps. Therefore, a cube is a regular polyhedron.
A polyhedron is semi-regular if all of its faces are regular polygons and the same number of faces meet at
every vertex.
•
Semi-regular polyhedra often have two different kinds of faces, both of which are regular polygons.
•
Prisms with a regular polygon base are one kind of semi-regular polyhedron.
•
Not all semi-regular polyhedra are prisms. An example of a non-prism is shown below.
Completely irregular polyhedra also exist. They are made of different kinds of regular and irregular polygons.
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So now a question arises. Given that a polyhedron is regular if all of its faces are congruent regular polygons,
it is convex and contains no gaps or holes. How many regular polyhedra actually exist?
In fact, you may be surprised to learn that only five regular polyhedra can be made. They are known as the
Platonic (or noble) solids.
Note that no matter how you try, you can’t construct any other regular polyhedra besides the ones above.
Example 8
How many faces, edges, and vertices does a tetrahedron (see above) have?
Faces: 4, edges: 6, vertices: 4
Example 9
Which regular polygon does an icosahedron feature?
An equilateral triangle
Review Exercises
Identify each of the following three-dimensional figures:
1.
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2.
3.
4.
5.
6. Below is a list of the properties of a polyhedron. Two of the properties are not correct. Find the incorrect
ones and correct them.
•
It is a 3 dimensional figure.
•
Some of its faces are polygons.
•
Polygon faces join together along segments called edges.
•
Each edge joins three faces.
•
There are no gaps between edges and vertices.
Complete the table and verify Euler’s formula for each of the figures in the problem.
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Figure
1. vertices
1. edges
1. faces
7. Pentagonal prism
8. Rectangular pyramid
9. Triangular prism
10. Trapezoidal prism
Answers
Identify each of the following three dimensional figures:
1. pentagonal prism
2. rectangular pyramid
3. triangular prism
4. triangular pyramid
5. trapezoidal prism
6. Below is a list of the properties of a polyhedron. Two of the properties are not correct. Find the incorrect
ones and correct them.
•
It is a 3 dimensional figure.
•
Some of its faces are polygons. All of its faces are polygons .
•
Polygon faces join together along segments called edges.
•
Each edge joins three faces. Each edge joins two faces .
•
There are no gaps between edges and vertices.
Complete the table and verify Euler’s formula for each of the figures in the problem.
Figure
1. vertices
1. edges
1. faces
7. Pentagonal prism
10
15
7
8. Rectangular pyramid
5
8
5
9. Triangular prism
6
9
5
10. Trapezoidal prism
8
12
6
In all cases vertices + faces = edges + 2
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Representing Solids
Learning Objectives
•
Identify isometric, orthographic, cross-sectional views of solids.
•
Draw isometric, orthographic, cross-sectional views of solids.
•
Identify, draw, and construct nets for solids.
Introduction
The best way to represent a three-dimensional figure is to use a solid model. Unfortunately, models are
sometimes not available. There are four primary ways to represent solids in two dimensions on paper. These
are:
•
An isometric (or perspective) view.
•
An orthographic or blow-up view.
•
A cross-sectional view.
•
A net.
Isometric View
The typical three-dimensional view of a solid is the isometric view. Strictly speaking, an isometric view of
a solid does not include perspective. Perspective is the illusion used by artists to make things in the distance
look smaller than things nearby by using a vanishing point where parallel lines converge.
The figures below show the difference between an isometric and perspective view of a solid.
As you can see, the perspective view looks more “real” to the eye, but in geometry, isometric representations
are useful for measuring and comparing distances.
The isometric view is often shown in a transparent “see-through” form.
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Color and shading can also be added to help the eye visualize the solid.
Example 1
Show isometric views of a prism with an equilateral triangle for its base.
Example 2
Show a see-through isometric view of a prism with a hexagon for a base.
Orthographic View
An orthographic projection is a blow-up view of a solid that shows a flat representation of each of the figure’s
sides. A good way to see how an orthographic projection works is to construct one. The (non-convex)
polyhedron shown has a different projection on every side.
To show the figure in an orthographic view, place it in an imaginary box.
Now project out to each of the walls of the box. Three of the views are shown below.
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A more complete orthographic blow-up shows the image of the side on each of the six walls of the box.
The same image looks like this in fold out view.
Example 3
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Show an orthographic view of the figure.
First, place the figure in a box.
Now project each of the sides of the figure out to the walls of the box. Three projections are shown.
You can use this image to make a fold-out representation of the same figure.
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Cross Section View
Imagine slicing a three-dimensional figure into a series of thin slices. Each slice shows a cross-section
view.
The cross section you get depends on the angle at which you slice the figure.
Example 4
What kind of cross section will result from cutting the figure at the angle shown?
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Example 5
What kind of cross section will result from cutting the figure at the angle shown?
Example 6
What kind of cross section will result from cutting the figure at the angle shown?
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Nets
One final way to represent a solid is to use a net. If you cut out a net you can fold it into a model of a figure.
Nets can also be used to analyze a single solid. Here is an example of a net for a cube.
There is more than one way to make a net for a single figure.
However, not all arrangements will create a cube.
Example 7
What kind of figure does the net create? Draw the figure.
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The net creates a box-shaped rectangular prism as shown below.
Example 8
What kind of net can you draw to represent the figure shown? Draw the net.
A net for the prism is shown. Other nets are possible.
Review Exercises
1. Name four different ways to represent solids in two dimensions on paper.
2. Show an isometric view of a prism with a square base.
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Given the following pyramid:
3. If the pyramid is cut with a plane parallel to the base, what is the cross section?
4. If the pyramid is cut with a plane passing through the top vertex and perpendicular to the base, what is
the cross section?
5. If the pyramid is cut with a plane perpendicular to the base but not through the top vertex, what is the
cross section?
Sketch the shape of the plane surface at the cut of this solid figure.
6. Cut AB
7. Cut CD
8. For this figure, what is the cross section?
Draw a net for each of the following:
9.
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10.
Answers
1. Name four different ways to represent solids in two dimensions on paper.
Isometric, orthographic, cross sectional, net
2. Show an isometric view of a prism with a square base.
Given the following pyramid:
3. If the pyramid is cut with a plane parallel to the base, what is the cross section? square
4. If the pyramid is cut with a plane passing through the top vertex and perpendicular to the base, what is
the cross section? triangle
5. If the pyramid is cut with a plane perpendicular to the base but not through the top vertex, what is the
cross section? trapezoid
Sketch the shape of the plane surface at the cut of this solid figure.
6.
7.
8. pentagon
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9.
10.
Prisms
Learning Objectives
•
Use nets to represent prisms.
•
Find the surface area of a prism.
•
Find the volume of a prism.
Introduction
A prism is a three-dimensional figure with a pair of parallel and congruent ends, or bases. The sides of a
prism are parallelograms. Prisms are identified by their bases.
Surface Area of a Prism Using Nets
The prisms above are right prisms. In a right prism, the lateral sides are perpendicular to the bases of
prism. Compare a right prism to an oblique prism, in which sides and bases are not perpendicular.
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Two postulates that apply to area are the Area Congruence Postulate and the Area Addition Postulate.
Area Congruence Postulate: If two polygons (or plane figures) are congruent, then their
areas are congruent.
Area Addition Postulate: The surface area of a three-dimensional figure is the sum of
the areas of all of its non-overlapping parts.
You can use a net and the Area Addition Postulate to find the surface area of a right prism.
From the net, you can see that that the surface area of the entire prism equals the sum of the figures that
make up the net:
Total surface area = area A + area B + area C + area D + area E + area F
Using the formula for the area of a rectangle, you can see that the area of rectangle A is:
square units
Similarly, the areas of the other rectangles are inserted back into the equation above.
Total surface area = area A + area B + area C + area D + area E + area F
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square units
Example 9
Use a net to find the surface area of the prism.
The area of the net is equal to the surface area of the figure. To find the area of the triangle, we use the
formula:
A = 1/2 hb where h is the height of the triangle and b is its base.
Note that triangles A and E are congruent so we can multiply the area of triangle A by 2.
Thus, the surface area is 324 square units.
Surface Area of a Prism Using Perimeter
This hexagonal prism has two regular hexagons for bases and six sides. Since all sides of the hexagon are
congruent, all of the rectangles that make up the lateral sides of the three-dimensional figure are also congruent. You can break down the figure like this.
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The surface area of the rectangular sides of the figure is called the lateral area of the figure. To find the
lateral area, you could add up all of the areas of the rectangles.
Notice that
is the perimeter of the base. So another way to find the lateral area of the figure is to multiply
the perimeter of the base by
, the height of the figure.
Substituting
, the perimeter, for
, we get the formula for any lateral area of a right prism:
Now we can use the formula to calculate the total surface area of the prism. Using
for the area of a base:
Total
area
for the perimeter and
surface = lateral area + area of 2 bases
= (perimeter of base • height) + 2 (area of base)
= Ph + 2 B
To find the surface area of the figure above, first find the area of the bases. The regular hexagon is made
of six congruent small triangles. The altitude of each triangle is the apothem of the polygon. Note: be
careful here—we are talking about the altitude of the triangles, not the height of the prism. We find the length
of the altitude of the triangle using the Pythagorean Theorem,
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So the area of each small triangle is:
The area of the entire hexagon is therefore:
You can also use the formula for the area of a regular polygon to find the area of each base:
Now just substitute values to find the surface area of the entire figure above.
You can use the formula
to find the surface area of any right prism.
Example 10
Use the formula to find the total surface area of the trapezoidal prism.
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The dimensions of the trapezoidal base are shown. Set up the formula. We’ll call the height of the entire
prism H to avoid confusion with h, the height of each trapezoidal base.
Total surface area
Now find the area of each trapezoidal base. You can do this by using the formula for the area of a trapezoid.
(Note that the height of the trapezoid, 2.46 is small h.)
Now find the perimeter of the base.
Now find the total surface area of the solid.
Volume of a Right Rectangular Prism
Volume is a measure of how much space a three-dimensional figure occupies. In everyday language, the
volume tells you how much a three-dimensional figure can hold. The basic unit of volume is the cubic
unit—cubic centimeter, cubic inch, cubic meter, cubic foot, and so on. Each basic cubic unit has a measure
of 1 for its length, width, and height.
Two postulates that apply to volume are the Volume Congruence Postulate and the Volume Addition Postulate.
Volume Congruence Postulate If two polyhedrons (or solids) are congruent, then their
volumes are congruent.
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Volume Addition Postulate The volume of a solid is the sum of the volumes of all of its
non-overlapping parts.
A right rectangular prism is a prism with rectangular bases and the angle between each base and its rectangular lateral sides is also a right angle. You can recognize a right rectangular prism by its “box” shape.
You can use the Volume Addition Postulate to find the volume of a right rectangular prism by counting boxes.
The box below measures 2 units in height, 4 units in width, and 3 units in depth. Each layer has 2 x 4 cubes
or 8 cubes.
Together, you get three groups of
so the total volume is:
The volume is 24 cubic units.
This same pattern holds for any right rectangular prism. Volume is giving by the formula:
Example 11
Find the volume of this box.
Use the formula for volume of a right rectangular prism.
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So the volume of this rectangular prism is
cubic units.
Volume of a Right Prism
Looking at the volume of right prisms with the same height and different bases you can see a pattern. The
computed area of each base is given below. The height of all three solids is the same,
.
Putting the data for each solid into a table, we get:
Solid
Height
Area of base
Volume
Box
10
300
3000
Trapezoid
10
140
1400
Triangle
10
170
1700
The relationship in each case is clear. This relationship can be proved to establish the following formula for
any right prism:
Volume of a Right Prism The volume of a right prism is
.
is the area of the base of the three-dimensional figure, and
where
altitude).
is the prism’s height (also called
Example 12
Find the volume of the prism with a triangular equilateral base and the dimensions shown in centimeters.
To find the volume, first find the area of the base. It is given by:
The height (altitude) of the triangle is 10.38 cm. Each side of the triangle measures 12 cm. So the triangle
has the following area.
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Now use the formula for the volume of the prism,
, where
is the area of the base (i.e., the
area of the triangle) and
is the height of the prism. Recall that the "height" of the prism is the distance
between the bases, so in this case the height of the prism is 15 cm. You can imagine that the prism is lying
on its side.
Thus, the volume of the prism is
Example 13
Find the volume of the prism with a regular hexagon for a base and 9-inch sides.
You don’t know the apothem of the figure’s base. However, you do know that a regular hexagon is divided
into six congruent equilateral triangles.
You can use the Pythagorean Theorem to find the apothem. The right triangle measures
the apothem.
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by
by
,
Thus, the volume of the prism is given by:
Review Exercises
For each of the following find the surface are a using
a. the method of nets and
b. the perimeter.
1.
2.
3. The base of a prism is a right triangle whose legs are 3 and 4 and show height is 20. What is the total
area of the prism?
4. A right hexagonal prism is 24 inches tall and has bases that are regular hexagons measuring 8 inches
on a side. What is the total surface area?
5. What is the volume of the prism in problem #4?
For problems 6 and 7:
A barn is shaped like a pentagonal prism with dimensions shown in feet:
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6. How many square feet (excluding the roof) are there on the surface of the barn to be painted?
7. If a gallon of paint covers 250 square feet, how many gallons of paint are needed to paint the barn?
8. A cardboard box is a perfect cube with an edge measuring 17 inches. How many cubic feet can it hold?
9. A swimming pool is 16 feet wide, 32 feet long and is uniformly 4 feet deep. How many cubic feet of water
can it hold?
10. A cereal box has length 25 cm, width 9 cm and height 30 cm. How much cereal can it hold?
Answers
2
1. 40.5 in
2
2. 838 cm
3. 252 square units
4. 1484.6 square units
5. 7981.3 cubic units
6. 2450 square feet
7. 10 gallons of paint
8. 2.85 cubic feet (be careful here. The units in the problem are given in inches but the question asks for
feet.)
9. 2048 cubic feet
3
10. 6750 cm
Cylinders
Learning Objectives
•
Find the surface area of cylinders.
•
Find the volume of cylinders.
•
Find the volume of composite three-dimensional figures.
Introduction
A cylinder is a three-dimensional figure with a pair of parallel and congruent circular ends, or bases. A
cylinder has a single curved side that forms a rectangle when laid out flat.
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As with prisms, cylinders can be right or oblique. The side of a right cylinder is perpendicular to its circular
bases. The side of an oblique cylinder is not perpendicular to its bases.
Surface Area of a Cylinder Using Nets
You can deconstruct a cylinder into a net.
The area of each base is given by the area of a circle:
The area of the rectangular lateral area L is given by the product of a width and height. The height is given
as 24. You can see that the width of the area is equal to the circumference of the circular base.
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To find the width, imagine taking a can-like cylinder apart with a scissors. When you cut the lateral area,
you see that it is equal to the circumference of the can’s top. The circumference of a circle is given by
the lateral area, L, is
Now we can find the area of the entire cylinder using
.
You can see that the formula we used to find the total surface area can be used for any right cylinder.
Area of a Right Cylinder The surface area of a right cylinder, with radius
is given by
, where
is the lateral area of the cylinder.
is the area of each base of the cylinder and
Example 1
Use a net to find the surface area of the cylinder.
First draw and label a net for the figure.
Calculate the area of each base.
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and height
Calculate
.
Find the area of the entire cylinder.
Thus, the total surface area is approximately 854.08 square units
Surface Area of a Cylinder Using a Formula
You have seen how to use nets to find the total surface area of a cylinder. The postulate can be broken
down to create a general formula for all right cylinders.
Notice that the base,
The lateral area,
, of any cylinder is:
, for any cylinder is:
Putting the two equations together we get:
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Factoring out a
from the equation gives:
The Surface Area of a Right Cylinder A right cylinder with radius
be expressed as:
or:
You can use the formulas to find the area of any right cylinder.
Example 2
Use the formula to find the surface area of the cylinder.
Write the formula and substitute in the values and solve.
Example 3
Find the surface area of the cylinder.
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and height
can
Write the formula and substitute in the values and solve.
Example 4
Find the height of a cylinder that has radius
cm and surface area of
sq cm.
Write the formula with the given information and solve for
.
Volume of a Right Cylinder
You have seen how to find the volume of any right prism.
where
is the area of the prism’s base and
is the height of the prism.
As you might guess, right prisms and right cylinders are very similar with respect to volume. In a sense, a
cylinder is just a “prism with round bases.” One way to develop a formula for the volume of a cylinder is to
compare it to a prism. Suppose you divided the prism above into slices that were unit thick.
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The volume of each individual slice would be given by the product of the area of the base and the height.
Since the height for each slice is , the volume of a single slice would be:
Now it follows that the volume of the entire prism is equal to the area of the base multiplied by the number
slices, then:
of slices. If there are
Of course, you already know this formula from prisms. But now you can use the same idea to obtain a formula
for the volume of a cylinder.
Since the height of each unit slice of the cylinder is
the base has an area of
, each slice has a volume of
, each slice has a volume of
and:
This leads to a postulate for the volume of any right cylinder.
Volume of a Right Cylinder The volume of a right cylinder with radius
can be expressed as:
Example 5
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and height
, or
. Since
Use the postulate to find the volume of the cylinder.
Write the formula from the postulate. Then substitute in the values and solve.
Example 6
What is the radius of a cylinder with height
cm and a volume of
?
Write the formula. Solve for
.
Composite Solids
Suppose this pipe is made of metal. How can you find the volume of metal that the pipe is made of?
The basic process takes three steps.
Step 1: Find the volume of the entire cylinder as if it had no hole.
Step 2: Find the volume of the hole.
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Step 3: Subtract the volume of the hole from the volume of the entire cylinder.
Here are the steps carried out. First, use the formula to find the volume of the entire cylinder. Note that since
, the diameter of the pipe, is cm, the radius is half of the diameter, or cm.
Now find the volume of the inner empty “hole” in the pipe. Since the pipe is
hole is inches less than the diameter of the outer part of the pipe.
The radius of the hole is half of
or
inch thick, the diameter of the
.
Now subtract the hole from the entire cylinder.
Example 7
Find the solid volume of this cinder block. Its edges are
cm thick all around. The two square holes are identical in size.
Find the volume of the entire solid block figure. Subtract the volume of the two holes.
To find the volume of the three-dimensional figure:
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Now find the length of the sides of the two holes. The width of the entire block is 21 cm. This is equal to:
So the sides of the square holes are 6 cm by 6 cm.
Now the volume of each square hole is:
Finally, subtract the volume of the two holes from the volume of the entire brick.
Review Exercises
Complete the following sentences. They refer to the figure above.
1. The figure above is a _________________________
2. The shape of the lateral face of the figure is _____________________________
3. The shape of a base is a(n) _____________________________
4. Segment LV is the ___________________________
5. Draw the net for this cylinder and use the net to find the surface area of the cylinder.
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6. Use the formula to find the volume of this cylinder.
7. Matthew’s favorite mug is a cylinder that has a base area of 9 square inches and a height of 5 inches.
How much coffee can he put in his mug?
8. Given the following two cylinders which of the following statements is true:
a. Volume of A < Volume of B
b. Volume of A > Volume of B
c. Volume of A = Volume of B
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9. Suppose you work for a company that makes cylindrical water tanks. A customer wants a tank that
measures 9 meters in height and 2 meters in diameter. How much metal should you order to make this
tank?
10. If the radius of a cylinder is doubled what effect does the doubling have on the volume of this cylinder?
Explain your answer.
Answers
1. Cylinder
2. Rectangle
3. Circle
4. Height
2
5. Surface area = 266π in
2
6. 250πcm
3
7. Volume = 45 in
8. Volume of A < volume of B
2
9. 18πm
10. The volume will be quadrupled
Pyramids
Learning Objectives
•
Identify pyramids.
•
Find the surface area of a pyramid using a net or a formula.
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•
Find the volume of a pyramid.
Introduction
A pyramid is a three-dimensional figure with a single base and a three or more non-parallel sides that meet
at a single point above the base. The sides of a pyramid are triangles.
A regular pyramid is a pyramid that has a regular polygon for its base and whose sides are all congruent
triangles.
Surface Area of a Pyramid Using Nets
You can deconstruct a pyramid into a net.
To find the surface area of the figure using the net, first find the area of the base:
Now find the area of each isosceles triangle. Use the Pythagorean Theorem to find the height of the triangles.
This height of each triangle is called the slant height of the pyramid. The slant height of the pyramid is the
altitude of one of the triangles. Notice that the slant height is larger than the altitude of the triangle.
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We’ll call the slant height
for this problem. Using the Pythagorean Theorem:
Now find the area of each triangle:
As there are
triangles:
Finally, add the total area of the triangles to the area of the base.
Example 1
Use the net to find the total area of the regular hexagonal pyramid with an apothem of
. The dimensions are given in centimeters.
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The area of the hexagonal base is given by the formula for the area of a regular polygon. Since each side
of the hexagon measures cm, the perimeter is
or
cm. The apothem, or perpendicular distance
to the center of the hexagon is
cm.
Using the Pythagorean Theorem to find the slant height of each lateral triangle.
Now find the area of each triangle:
Together, the area of all six triangles that make up the lateral sides of the pyramid are
Add the area of the lateral sides to the area of the hexagonal base.
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Surface Area of a Regular Pyramid
To get a general formula for the area of a regular pyramid, look at the net for this square pyramid.
The slant height of each lateral triangle is labeled (the lowercase letter
polygon is labeled . For each lateral triangle, the area is:
There are
pyramid,
), and the side of the regular
triangles in a regular polygon—e.g.,
for a triangular pyramid,
for a pentagonal pyramid. So the total area,
, of the lateral triangles is:
for a square
If we rearrange the above equation we get:
Notice that
is just the perimeter,
to get the following postulate.
, of the regular polygon. So we can substitute
To get the total area of the pyramid, add the area of the base,
into the equation
, to the equation above.
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Area of a Regular Pyramid The surface area of a regular pyramid is
where is the slant height of the pyramid and
pyramid’s base, and
is the area of the base.
is the perimeter of the regular polygon that forms the
Example 2
A tent without a bottom has the shape of a hexagonal pyramid with a slant height
of
feet. The sides of the hexagonal perimeter of the figure each measure
feet. Find the surface area of the tent that exists above ground.
For this problem,
figure.
is zero because the tent has no bottom. So simply calculate the lateral area of the
Example 3
A pentagonal pyramid has a slant height
of
cm. The sides of the pentagonal perimeter of the figure each measure
cm. The apothem of the figure is
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cm. Find the surface area of the figure.
First find the lateral area of the figure.
Now use the formula for the area of a regular polygon to find the area of the base.
Finally, add these together to find the total surface area.
Estimate the Volume of a Pyramid and Prism
Which has a greater volume, a prism or a pyramid, if the two have the same base and height? To find out,
compare prisms and pyramids that have congruent bases and the same height.
Here is a base for a triangular prism and a triangular pyramid. Both figures have the same height. Compare
the two figures. Which one appears to have a greater volume?
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The prism may appear to be greater in volume. But how can you prove that the volume of the prism is greater
than the volume of the pyramid? Put one figure inside of the other. The figure that is smaller will fit inside of
the other figure.
This is shown in the diagram on the above. Both figures have congruent bases and the same height. The
pyramid clearly fits inside of the prism. So the volume of the pyramid must be smaller.
Example 4
Show that the volume of a square prism is greater than the volume of a square pyramid.
Draw or make a square prism and a square pyramid that have congruent bases and the same height.
Now place the one figure inside of the other. The pyramid fits inside of the prism. So when two figures have
the same height and the same base, the prism’s volume is greater.
In general, when you compare two figures that have congruent bases and are equal in height, the prism will
have a greater volume than the pyramid.
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The reason should be obvious. At the “bottom,” both figures start out the same—with a square base. But
the pyramid quickly slants inward, “cutting away” large amounts of material while the prism does not slant.
Find the Volume of a Pyramid and Prism
Given the figure above, in which a square pyramid is placed inside of a square prism, we now ask: how
many of these pyramids would fit inside of the prism?
To find out, obtain a square prism and square pyramid that are both hollow, both have no bottom, and both
have the same height and congruent bases.
Now turn the figures upside down. Fill the pyramid with liquid. How many full pyramids of liquid will fill the
prism up to the top?
In fact, it takes exactly three full pyramids to fill the prism. Since the volume of the prism is:
where
stands for the area of the base and
is the height of the prism, we can write:
or:
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And, since the volume of a square prism is
, we can write:
This can be written as the Volume Postulate for pyramids.
Volume of a Pyramid Given a right pyramid with a base that has area
:
and height
Example 5
Find the volume of a pyramid with a right triangle base with sides that measure
cm,
cm, and
cm. The height of the pyramid is
cm.
First find the area of the base. The longest of the three sides that measure cm,
be the hypotenuse, so the two shorter sides are the legs of the right triangle.
cm, and
Now use the postulate for the volume of a pyramid.
Example 6
Find the altitude of a pyramid with a regular pentagonal base. The figure has an apothem of
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cm must
cm,
cm sides, and a volume of
cu cm.
First find the area of the base.
Now use the value for the area of the base and the postulate to solve for
.
Review Exercises
Consider the following figure in answering questions 1 – 4.
1. What type of pyramid is this?
2. Triangle ABE is what part of the figure?
3. Segment AE is a(n) _______________ of the figure.
4. Point E is the ________________________
5. How many faces are there on a pyramid whose base has 16 sides?
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A right pyramid has a regular hexagon for a base. Each edge measures 2√22. Find
6. The lateral surface area of the pyramid
7. The total surface area of the pyramid
8. The volume of the pyramid
9. The Transamerica Building in San Francisco is a pyramid. The length of each edge of the square base
is 149 feet and the slant height of the pyramid is 800 feet. What is the lateral area of the pyramid? How tall
is the building?
10. Given the following pyramid:
3
With c=22 mm, b=17 mm and volume =1433.67 mm what is the value of a?
Answers
1. Rectangular pyramid
2. Lateral face
3. Edge
4. Apex
5. 16
6. 135.6 square units
7. 200.55 square units
8. 171.84 cubic units
9. Lateral surface area = 238,400 square feet
Height = 796.5 feet
10. A = 11.5 mm
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Cones
Learning Objectives
•
Find the surface area of a cone using a net or a formula.
•
Find the volume of a cone.
Introduction
A cone is a three-dimensional figure with a single curved base that tapers to a single point called an apex.
The base of a cone can be a circle or an oval of some type. In this chapter, we will limit the discussion to
circular cones. The apex of a right cone lies above the center of the cone’s circle. In an oblique cone, the
apex is not in the center.
The height of a cone,
, is the perpendicular distance from the center of the cone’s base to its apex.
Surface Area of a Cone Using Nets
Most three-dimensional figures are easy to deconstruct into a net. The cone is different in this regard. Can
you predict what the net for a cone looks like? In fact, the net for a cone looks like a small circle and a sector,
or part of a larger circle.
The diagrams below show how the half-circle sector folds to become a cone.
Note that the circle that the sector is cut from is much larger than the base of the cone.
Example 1
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Which sector will give you a taller cone—a half circle or a sector that covers three-quarters of a circle? Assume that both sectors are cut from congruent circles.
Make a model of each sector.
The half circle makes a cone that has a height that is about equal to the radius of the semi-circle.
The three-quarters sector gives a cone that has a wider base (greater diameter) but its height as not as
great as the half-circle cone.
Example 2
Predict which will be greater in height—a cone made from a half-circle sector or a cone made from a onethird-circle sector. Assume that both sectors are cut from congruent circles.
The relationship in the example above holds true—the greater (in degrees) the sector, the smaller in height
of the cone. In other words, the fraction
greater height than a half sector.
is less than
, so a one-third sector will create a cone with
Example 3
Predict which will be greater in diameter—a cone made from a half-circle sector or a cone made from a
one-third-circle sector. Assume that the sectors are cut from congruent circles
Here you have the opposite relationship—the larger (in degrees) the sector, the greater the diameter of the
cone. In other words,
than a one-third sector.
is greater than
, so a one-half sector will create a cone with greater diameter
Surface Area of a Regular Cone
The surface area of a regular pyramid is given by:
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where
is the slant height of the figure,
Imagine a series of pyramids in which
is the perimeter of the base, and
is the area of the base.
, the number of sides of each figure’s base, increases.
As you can see, as
increases, the figure more and more resembles a circle. So in a sense, a circle approaches a polygon with an infinite number of sides that are infinitely small.
In the same way, a cone is like a pyramid that has an infinite number of sides that are infinitely small in
length.
Given this idea, it should come as no surprise that the formula for finding the total surface area of a cone is
, the
similar to the pyramid formula. The only difference between the two is that the pyramid uses
, the circumference of the base.
perimeter of the base, while a cone uses
Surface Area of a Right Cone The surface area of a right cone is given by:
Since the circumference of a circle is
:
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You can also express
as
to get:
Any of these forms of the equation can be used to find the surface area of a right cone.
Example 4
Find the total surface area of a right cone with a radius of
cm and a slant height of
cm.
Use the formula:
Example 5
Find the total surface area of a right cone with a radius of
feet and an altitude (not slant height) of
feet.
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Use the Pythagorean Theorem to find the slant height:
Now use the area formula.
Volume of a Cone
Which has a greater volume, a pyramid, cone, or cylinder if the figures have bases with the same “diameter”
(i.e., distance across the base) and the same altitude? To find out, compare pyramids, cylinders, and cones
that have bases with equal diameters and the same altitude.
Here are three figures that have the same dimensions—cylinder, a right regular hexagonal pyramid, and a
right circular cone. Which figure appears to have a greater volume?
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It seems obvious that the volume of the cylinder is greater than the other two figures. That’s because the
pyramid and cone taper off to a single point, while the cylinder’s sides stay the same width.
Determining whether the pyramid or the cone has a greater volume is not so obvious. If you look at the
bases of each figure you see that the apothem of the hexagon is congruent to the radius of the circle. You
can see the relative size of the two bases by superimposing one onto the other.
From the diagram you can see that the hexagon is slightly larger in area than the circle. So it follows that
the volume of the right hexagonal regular pyramid would be greater than the volume of a right circular cone.
And indeed it is, but only because the area of the base of the hexagon is slightly greater than the area of
the base of the circular cone.
The formula for finding the volume of each figure is virtually identical. Both formulas follow the same basic
form:
Since the base of a circular cone is, by definition, a circle, you can substitute the area of a circle,
the base of the figure. This is expressed as a volume postulate for cones.
Volume of a Right Circular Cone Given a right circular cone with height
that has radius :
Example 6
Find the volume of a right cone with a radius of
cm and a height of
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and a base
for
cm.
Use the formula:
Example 7
Find the volume of a right cone with a radius of
feet and a slant height of
feet.
Use the Pythagorean theorem to find the height:
Now use the volume formula.
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Review Exercises
1. Find the surface area of
2. Find the surface area of
2
3. Find the surface area of a cone with a height of 4 m and a base area of 281.2 m
In problems 4 and 5 find the missing dimension. Round to the nearest tenth of a unit.
4. Cone: volume = 424 cubic meters
Diameter = 18 meters
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Height = ________
2
5. Cone: surface area = 153.5 in
Radius = 4 inches
Slant height = ________
6. A cone shaped paper cup is 8 cm high with a diameter of 5 cm. If a plant needs 240 ml of water, about
how many paper cups of water are needed to water it? (1 mL = 1 cubic cm)
In problems 7 and 8 refer to this diagram. It is a diagram of a yogurt container. The yogurt container is a
truncated cone.
7. What is the surface area of the container?
8. What is the volume of the container?
3
9. Find the height of a cone that has a radius of 2 cm and a volume of 23 cm
3
10. A cylinder has a volume of 2120.6 cm and a base radius of 5 cm. What is the volume of a cone with
the same height but a base radius of 3 cm?
Answers
1. 483.8 square units
2. 312.6 square units
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2
3. Surface area = 75.4 m
4. Height = 5 meters
5. Slant height = 8 inches
6. 1.2 cups (approximately)
2
7. Surface area of the container = 152.62 cm
3
8. The volume of the container = 212.58 cm
9. Height = 5.49 cm
3
10. Volume of the cone = 254.45 cm
Spheres
Learning Objectives
•
Find the surface area of a sphere.
•
Find the volume of a sphere.
Introduction
A sphere is a three-dimensional figure that has the shape of a ball.
Spheres can be characterized in three ways.
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•
A sphere is the set of all points that lie a fixed distance
•
A sphere is the surface that results when a circle is rotated about any of its diameters.
from a single center point
.
•
A sphere results when you construct a polyhedron with an infinite number of faces that are infinitely
small. To see why this is true, recall regular polyhedra.
As the number of faces on the figure increases, each face gets smaller in area and the figure comes more
to resemble a sphere. When you imagine figure with an infinite number of faces, it would look like (and be!)
a sphere.
Parts of a Sphere
As described above, a sphere is the surface that is the set of all points a fixed distance from a center point
. Terminology for spheres is similar for that of circles.
The distance from
to the surface of the sphere is
, the radius.
The diameter, , of a sphere is the length of the segment connecting any two points on the sphere’s surface
. Note that you can find a diameter (actually an infinite number of diameters) on
and passing through
any plane within the sphere. Two diameters are shown in each sphere below.
A chord for a sphere is similar to the chord of a circle except that it exists in three dimensions. Keep in mind
that a diameter is a kind of chord—a special chord that intersects the center of the circle or sphere.
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A secant is a line, ray, or line segment that intersects a circle or sphere in two places and extends outside
of the circle or sphere.
A tangent intersects the circle or sphere at only one point.
In a circle, a tangent is perpendicular to the radius that meets the point where the tangent intersects with
the circle. The same thing is true for the sphere. All tangents are perpendicular to the radii that intersect
with them.
Finally, a sphere can be sliced by an infinite number of different planes. Some planes include point
center of the sphere. Other points do not include the center.
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, the
Surface Area of a Sphere
You can infer the formula for the surface area of a sphere by taking measurements of spheres and cylinders.
Here we show a sphere with a radius of and a right cylinder with both a radius and a height of and express
the area in terms of
.
Now try a larger pair, expressing the surface area in decimal form.
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Look at a third pair.
Is it a coincidence that a sphere and a cylinder whose radius and height are equal to the radius of the sphere
have the exact same surface area? Not at all! In fact, the ancient Greeks used a method that showed that
).
the following formula can be used to find the surface area of any sphere (or any cylinder in which
The Surface Area of a Sphere is given by:
Example 1
Find the surface area of a sphere with a radius of 14 feet.
Use the formula.
Example 2
Find the surface area of the following figure in terms of
.
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The figure is made of one half sphere or hemisphere, and one cylinder without its top.
Now find the area of the cylinder without its top.
Thus, the total surface area is
Volume of a Sphere
A sphere can be thought of as a regular polyhedron with an infinite number of congruent polygon faces. A
series polyhedra with an increasing number of faces is shown.
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As
, the number of faces increases to an infinite number, the figure approaches becoming a sphere.
So a sphere can be thought of as a polyhedron with an infinite number faces. Each of those faces is the
, the center of the sphere. This is shown below.
base of a pyramid whose vertex is located at
Each of the pyramids that make up the sphere would be congruent to the pyramid shown. The volume of
this pyramid is given by:
To find the volume of the sphere, you simply need to add up the volumes of an infinite number of infinitely
small pyramids.
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Now, it should be obvious that the sum of all of the bases of the pyramids is simply the surface area of the
sphere. Since you know that the surface area of the sphere is
the equation above.
, you can substitute this quantity into
Finally, as
increases and the surface of the figure becomes more “rounded,”
pyramid becomes equal to , the radius of the sphere. So we can substitute for
, the height of each
. This gives:
We can write this as a formula.
Volume of a Sphere Given a sphere that has radius
Example 3
Find the volume of a sphere with radius
m.
Use the postulate above.
Example 4
A sphere has a volume of
. Find its diameter.
Use the postulate above. Convert
to an improper fraction,
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Since
, the diameter is 8 units.
Review Exercises
3
1. Find the radius of the sphere that has a volume of 335 cm .
Determine the surface area and volume of the following shapes:
2.
3.
4. The radius of a sphere is 4. Find its volume and total surface area.
5. A sphere has a radius of 5. A right cylinder, having the same radius has the same volume. Find the height
and total surface area of the cylinder.
In problems 6 and 7 find the missing dimension.
3
6. Sphere: volume = 296 cm
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Diameter = ________
2
7. Sphere: surface area is 179 in .
Radius = _______
8. Tennis balls with a diameter of 3.5 inches are sold in cans of three. The can is a cylinder. Assume the
balls touch the can on the sides, top and bottom. What is the volume of the space not occupied by the tennis
balls?
2
9. A sphere has surface area of 36π in . Find its volume.
10. A giant scoop, operated by a crane, is in the shape of a hemisphere of radius = 21 inches. The scoop
is filled with melted hot steel. When the steel is poured into a cylindrical storage tank that has a radius of
28 inches, the melted steel will rise to a height of how many inches?
Answers
1. Radius = 4.39 cm
2
2. Surface area = 706.86 cm
3
Volume = 1767.15 cm
2
3. Surface area = surface area of hemisphere + surface area of cone = 678.58 in
3
Volume = 2544.69 in
3
4. Volume = 268.08 units
2
Surface area = 201.06 units
2
5. Height = 20/3 units total surface area = 366.52 units
6. Diameter = 8.27 inches
7. Radius = 3.77 inches
3
8. Volume of cylinder = 32.16π in3 volume of tennis balls = 21.44π in
3
Volume of space not occupied by tennis balls = 33.67 in
3
9. Volume = 113.10 in
10. Height of molten steel in cylinder will be 7.88 inches
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Similar Solids
Learning Objectives
•
Find the volumes of solids with bases of equal areas.
Introduction
You’ve learned formulas for calculating the volume of different types of solids—prisms, pyramids, cylinders,
and spheres. In most cases, the formulas provided had special conditions. For example, the formula for the
volume of a cylinder was specific for a right cylinder.
Now the question arises: What happens when you consider the volume of two cylinders that have an equal
base but one cylinder is non-right—that is, oblique. Does an oblique cylinder have the same volume as a
right cylinder if the two share bases of the same area?
Parts of a Solid
Given, two cylinders with the same height and radius. One cylinder is a right cylinder, the other is oblique.
To see if the volume of the oblique cylinder is equivalent to the volume of the right cylinder, first observe the
two solids.
Since they both have the same circular radius, they both have congruent bases with area:
Now cut the right cylinder into a series of
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cross-section disks each with height
and radius
.
It should be clear from the diagram that the total volume of the
cylinder.
disks is equal to the volume of the original
Now start with the same set of disks. Shift each disk over to the right. The volume of the shifted disks must
be exactly the same as the unshifted disks, since both figures are made out of the same disks.
It follows that the volume of the oblique figure is equal to the volume of the original right cylinder.
In other words, if the radius and height of each figure are congruent:
The principle shown above was developed in the seventeenth century by Italian mathematician Francisco
Cavalieri. It is known as Cavalieri’s Principle. (Liu Hui also discovered the same principle in third-century
China, but was not given credit for it until recently.) The principle is valid for any solid studied in this chapter.
Volume of a Solid Postulate (Cavalieri’s Principle):
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The volumes of two objects are equal if the areas of their corresponding cross-sections are in all cases
equal. Two cross-sections correspond if they are intersections of the figure with planes equidistant from a
chosen base plane.
Example 1
Prove (informally) that the two circular cones with the same radius and height are equal in volume.
As before, we can break down the right circular cone into disks.
Now shift the disks over.
You can see that the shifted-over figure, since it uses the very same disks as the straight figure, must have
the same volume. In fact, you can shift the disks any way you like. Since you are always using the same
set of disks, the volume is the same.
Keep in mind that Cavalieri’s Principle will work for any two solids as long as their bases are equal in area
(not necessarily congruent) and their cross sections change in the same way.
Example 2
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A rectangular pyramid and a circular cone have the same height, and base area. Are their volumes congruent?
Yes. Even though the two figures are different, both can be computed by using the following formula:
Since
Then
Similar or Not Similar
Two solids of the same type with equal ratios of corresponding linear measures (such as heights or radii)
are called similar solids.
To be similar, figures need to have corresponding linear measures that are in proportion to one another. If
these linear measures are not in proportion, the figures are not similar.
Example 1
Are these two figures similar?
If the figures are similar, all ratios for corresponding measures must be the same.
The ratios are:
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Since the three ratios are equal, you can conclude that the figures are similar.
Example 2
Cone A has height
and radius
. Cone B has height
and radius
. Are the two cones similar?
If the figures are similar all ratios for corresponding measures must be the same.
The ratios are:
Since the ratios are different, the two figures are not similar.
Compare Surface Areas and Volumes of Similar Figures
When you compare similar two-dimensional figures, area changes as a function of the square of the ratio
of
For example, take a look at the areas of these two similar figures.
The ratio between corresponding sides is:
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The ratio between the areas of the two figures is the square of the ratio of the linear measurement:
This relationship holds for solid figures as well. The ratio of the areas of two similar figures is equal to the
square of the ratio between the corresponding linear sides.
Example 3
Find the ratio of the surface area between the two similar figures C and D.
Since the two figures are similar, you can use the ratio between any two corresponding measurements to
find the answer. Here, only the radius has been supplied, so:
The ratio between the areas of the two figures is the square of the ratio of the linear measurements:
Example 4
If the surface area of the small cylinder in the problem above is 80π, what is the surface area of the larger
cylinder?
From above we, know that:
So the surface area can be found by setting up equal ratios
Solve for n.
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The ratio of the volumes of two similar figures is equal to the cube of the ratio between the corresponding
linear sides.
Example 5
Find the ratio of the volume between the two similar figures C and D.
As with surface area, since the two figures are similar you can use the height, depth, or width of the figures
to find the linear ratio. In this example we will use the widths of the two figures.
The ratio between the volumes of the two figures is the cube of the ratio of the linear measurements:
Does this cube relationship agree with the actual measurements? Compute the volume of each figure.
As you can see, the ratio holds. We can summarize the information in this lesson in the following postulate.
Similar Solids Postulate: If two solid figures, A and B are similar and the ratio of their
linear measurements is
, then the ratio of their surface areas is:
The ratio of their volumes is:
Scale Factors and Models
The ratio of the linear measurements between two similar figures is called the scaling factor. For example,
we can find the scaling factor for cylinders E and F by finding the ratio of any two corresponding measurements.
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Using the heights, we find a scaling factor of:
.
You can use a scaling factor to make a model.
Example 6
Doug is making a model of the Statue of Liberty. The real statue has a height of
feet and a nose that is
feet in length. Doug’s model statue has a height of
feet. How long should the nose on Doug’s model be?
First find the scaling factor.
To find the length of the nose, simply multiply the height of the model’s nose by the scaling factor.
In inches, the quantity would be:
Example 7
An architect makes a scale model of a building shaped like a rectangular prism. The model measures
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ft in height,
inches in width, and
inches in depth. The real building will be
feet tall. How wide will the real building be?
First find the scaling factor.
To find the width, simply multiply the width of the model by the scaling factor.
Review Exercises
1. How does the volume of a cube change if the sides of a cube are multiplied by 4? Explain.
2. In a cone if the radius and height are doubled what happens to the volume? Explain.
3. In a rectangular solid, is the sides are doubled what happens to the volume? Explain.
4. Two spheres have radii of 5 and 9. What is the ratio of their volumes?
5. The ratio of the volumes of two similar pyramids is 8:27. What is the ratio of their total surface areas?
6. A) Are all spheres similar? B) Are all cylinders similar? C) Are all cubes similar? Explain your answers to
each of these.
7. The ratio of the volumes of two tetrahedron is 1000:1. The smaller tetrahedron has a side of length 6
centimeters. What is the side length of the larger tetrahedron?
Refer to these two similar cylinders in problems 8 – 10:
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8. What is the similarity ratio of cylinder A to cylinder B?
9. What is the ratio of surface area of cylinder A to cylinder B?
10. What is the ratio of the volume of cylinder B to cylinder A?
Answers
3
3
1. The volume will be 64 times greater. Volume = s New volume = (4s)
2. Volume will be 8 times greater.
3. The volume will be 8 times greater (2w)(2l)(2h) = 8 wlh = 8 (volume of first rectangular solid)
3
3
4. 5 /9
5. 4/9
6. All spheres and all cubes are similar since each has only one linear measure. All cylinders are not similar.
They can only be similar if the ratio of the radii = the ratio of the heights.
7. 60 cm
8. 20/5 = 4/1
9. 16/1
3
10. 1/4
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